differs by signs from the formulation in \cref{ONH1,ONH2,ONH3,ONH4,ONH5,ONH6}.
This discrepancy is explained by the signs in formula \cref{ukraine2}.
To be clear about this,
for $a \in \ONH_d$,
let $\rho_{(1^d);n}(a)$ be the
endomorphism from \cref{out} viewed now as an endomorphism
of the degree-shifted $Q^{-nd - \binom{d}{2}} U^\ell_{n+d-1}\otimes_{\EOH_{n+d-1}^\ell}
\cdots\:\:\otimes_{\EOH_{n+1}^\ell}  U^\ell_{n}$,
for $0 \leq d \leq n$ and $0 \leq n \leq \ell-d$.
The definition from (3) implies more generally that
\begin{align}\label{inhand1}
\Psi_\ell\left(
\begin{tikzpicture}[anchorbase,scale=.8]
\draw[->] (1,-.3) to (1,.4);
\node at (.5,.05) {$\cdots$};
\node at (-.5,.05) {$\cdots$};
\draw[->] (-1,-.3) to (-1,.4);
\draw[->] (0.0,-.3) to (0.0,.4);
\node at (0,-.5) {$\stringnumber{i}$};
\node at (1,-.5) {$\stringnumber{1}$};
\node at (-1,-.5) {$\stringnumber{d}$};
\opendot{0.0,0.05};
\node at (1.5,.1) {$\red\scriptstyle{2n-\ell}$};
\end{tikzpicture}
\right)
&= (-1)^{i-1} \rho_{(1^d);n}(x_i),\\\label{inhand2}
\Psi_\ell\left(
\begin{tikzpicture}[anchorbase,scale=.8]
\draw[->] (1,-.3) to (1,.4);
\node at (.5,.05) {$\cdots$};
\node at (-.8,.05) {$\cdots$};
\draw[->] (-1.3,-.3) to (-1.3,.4);
\draw[->] (0.0,-.3) to (-0.3,.4);
\draw[->] (-.3,-.3) to (0.0,.4);
\node at (0.03,-.5) {$\stringnumber{j}$};
\node at (-0.33,-.5) {$\stringnumber{j+1}$};
\node at (1,-.5) {$\stringnumber{1}$};
\node at (-1.3,-.5) {$\stringnumber{d}$};
\node at (1.5,.1) {$\red\scriptstyle{2n-\ell}$};
\end{tikzpicture}
\right)
&= -(-1)^{j-1} \rho_{(1^d);n}(\tau_j).
\end{align}
We check \cref{inhand2}, leaving the easier \cref{inhand1} to the reader.
We must show that
\begin{equation*}
\id\otimes\cdots\otimes \rho_{(1^2);n+j-1}(\tau_1)\otimes\cdots\otimes \id
= (-1)^{j-1} \rho_{(1^d);n}(\tau_j).
\end{equation*}
We do this by checking that both sides take the same value on
$u_{n+d-1}\big(x^{\kappa_d}\big)\otimes\cdots \otimes u_n\big(x^{\kappa_1}\big)$
for any $\kappa \in \N^d$.
Let $\sum_{\kappa'}$ denote summation over
$\kappa'\in\N^d$ with $\kappa_i' = \kappa_i$ for $i \neq j,j+1$.
Suppose that
$x_{j+1}^{\kappa_{j+1}} x_j^{\kappa_j}
\cdot \tau_j
= \sum_{\kappa'} c_{\kappa'}x_{j+1}^{\kappa_{j+1}'} x_j^{\kappa_j'}$.
Then, using \cref{ukraine2}, the left hand side gives
\sum_{\kappa'}
u_{n+d-1}\big(x^{\kappa_d'}\big)\otimes\cdots \otimes
and the right hand side gives
(-1)^{d-1+(d-j)(\kappa_j - \kappa_j')+
(d-j-1)(\kappa_{j+1}-\kappa_{j+1}') +
\kappa_{j}+\cdots+\kappa_d} c_{\kappa'}
\otimes u_n\big(x^{\kappa'_1}\big).
These are equal because
$\kappa_j - \kappa_j'+\kappa_{j+1}-\kappa_{j+1}'=1$
whenever $c_{\kappa'} \neq 0$.

With \cref{inhand1,inhand2} in hand,  the relations \cref{nearlydone} are easily checked.
For example, to check the length three braid relation,
we must show that
\rho_{(1^3);n}(\tau_2) \circ
\big(-\rho_{(1^3);n}(\tau_1)\big) \circ
\rho_{(1^3);n}(\tau_2)
\big(-\rho_{(1^3);n}(\tau_1)\big) \circ \rho_{(1^3);n}(\tau_2) \circ \big(-\rho_{(1^3);n}(\tau_1)\big).
Translating to $U_{(1^3);n}^\ell$
using the isomorphism $b_{(1)^3}$, the
left hand side becomes the map $u_{(1^3);n}(f) \mapsto (-1)^{\parity(f)}
u_{(1^3);n}(f \cdot \tau_2 \tau_1 \tau_2)$
and the right hand side becomes
$u_{(1^3);n}(f) \mapsto -(-1)^{\parity(f)}
u_{(1^3);n}(f \cdot \tau_1 \tau_2 \tau_1)$.
These are equal due to the sign in the
relation \cref{ONH4}.
To check the third relation in \cref{nearlydone},
we must show that
\rho_{(1^2);n}(x_1)\circ \big(-\rho_{(1^2);n}(\tau_1)\big)+
\rho_{(1^2);n}(\tau_1) \circ \rho_{(1^2);n}(x_2)
= \rho_{(1^2);n}(\id).
The left hand side corresponds to the map
$u_{(1^2);n}(f) \mapsto u_{(1^2);n}(f \cdot \tau_1 x_1 -f \cdot x_2 \tau_1)$,
which equals $u_{(1^2);n}(f)$ by \cref{ONH6}.

Next we check \cref{inv1,inv2}.
There is nothing to do if $\ell = 0$ (the zero map is an isomorphism between zero superbimodules!) so assume that $\ell > 0$.
Take a weight $k=2n-\ell$ of $\mathbf{V}(-\ell)$
for some $0 \leq n \leq \ell$.
Setting $n' := \ell-n-1$, we have that $k = n-n'-1$.
For brevity, we will ignore grading shifts since they play no role in this place, i.e., we work with ordinary rather than
graded superbimodules.

We first check \cref{inv1}, so  $k \geq 0$ or, equivalently,
$n \geq n'+1$.
We need to show that the superbimodule homomorphism $f$
defined by the $(n-n')\times 1$ matrix
\left(\:
\sigma_n
\quad
\ev_{n-1}
\quad
%\ev_{n-1} \circ \rho^{(2)}_n(x_2)
\cdots
\quad
\ev_{n-1} \circ \big(\rho_{(1);n-1}(x)^{n-n'-2} \otimes \id\big)
\right)^T\!:
U_{n-1} \otimes_{\EOH_{n-1}^\ell} \!V_{n-1}^\ell
\rightarrow
V_n^\ell\otimes_{\EOH_{n+1}^\ell} U_n^\ell
\oplus (\EOH_n^\ell)^{\oplus (n-n'-1)}
is an isomorphism where $\sigma_n$, the image of the rightward crossing,
is the superbimodule homomorphism described explicitly in \cref{mate2},
or the zero map in the extremal case $n=\ell, n'=-1$.
By \cref{lemma3},
the domain of $f$ is free as a right $\EOH_n^\ell$-supermodule
with basis
$\big\{u_{n-1}(x^r) \otimes v_{n-1}(x^s)\:\big|\:
0 \leq r \leq n'+1, 0 \leq s \leq n-1\big\}$,
and the codomain of $f$ is
free as a right $\EOH_n^\ell$-supermodule with basis
$\big\{v_n(x^s)\otimes u_n(x^r)\:\big|\:0 \leq r \leq n',
0 \leq s \leq n\big\} \cup \big\{b_1,\dots,b_{n-n'-1}\big\}$
where $b_i$ is the identity element in the $i$th copy of $\EOH_n^\ell$.
Both of these sets are of size $nn'+2n$, so it suffices
to show that $f$ is {\em surjective}.
To prove this, since $\EOH_n^\ell$ is graded local,
it is enough to show that the homomorphism
$\bar f := f \otimes 1$
obtained by applying the functor $-\otimes_{\EOH_n^\ell} \k$
is surjective.
Let $uv(r,s)$ denote
$u_{n-1}(x^r)\otimes v_{n-1}(x^s) \otimes 1$ and
$vu(s,r)$ denote $v_n(x^s) \otimes u_n(x^r) \otimes 1$.
Thus, the domain of $\bar f$ has linear basis
0 \leq r \leq n'+1, 0 \leq s \leq n-1\}$
and the codomain has linear basis
$\{vu(s,r)\:|\:0 \leq r \leq n',
0 \leq s \leq n\} \cup \{b_1\otimes 1,\dots, b_{n-n'-1}\otimes 1\}$.
By \cref{Ian} and \cref{cupsandcaps}, we have that
\begin{equation}\label{howitgoes}
\bar f\big(uv(r,s)\big) =
\pm vu(n,r+s-n)
\pm b_{n-r-s}\otimes 1
\pm vu(s,r)
\end{equation}
for $0 \leq r\leq n'+1$ and $0 \leq s \leq n-1$,
where the first term should be interpreted as zero if $r+s-n < 0$
and the middle term should be interpreted as zero
if $n-r-s < 1$ or $n-r-s > n-n'-1$.
Note also that the last term $vu(s,r)$ is zero for $r > n'$ by degree considerations.
The argument is completed with the following observations.
\begin{itemize}
\item
We get the vectors
$vu(n,r)$ for $0 \leq r \leq n'$ from the images of
the basis vectors $uv(n'+1,s)$ for $n-n'-1\leq s \leq n-1$.
Indeed, in the formula \cref{howitgoes}
for $\bar f(uv(n'+1,s))$ for these values of $s$,
the second and third terms on the right hand side are both zero.
\item
Modulo the span of vectors already obtained,
we get the vectors $b_1\otimes 1,\dots, b_{n-n'-1}\otimes 1$
from the images of the basis vectors
$uv(n'+1,s)$ for $0 \leq s \leq n-n'-2$.
Indeed, in the formula for $f(uv(n'+1,s))$ for these values of $s$
the third term on the right hand side is zero.
\item
Modulo the span of vectors already obtained,
we get the vectors
for $0 \leq r \leq n'$ and $0 \leq s \leq n-1$
from the images of the remaining basis vectors $uv(r,s)$
for these values of $r$ and $s$.
\end{itemize}

Now consider \cref{inv2}, so $k \leq 0$ and $n'\geq n-1$.
We need to show that the superbimodule homomorphism $f$
defined by the $1\times (n'-n+2)$ matrix
\left(\:
\sigma_n
\quad
\coev_n
%\rho^{(2)}_n(x_1) \circ \coev_n
\quad\cdots\quad
\big(\id \otimes \rho_{(1);n}(x)^{n'-n}\big) \circ \coev_n
\right):
U_{n-1} \otimes_{\EOH_{n-1}^\ell}\! V_{n-1}^\ell
\oplus (\EOH_n^\ell)^{\oplus (n'-n+1)}\rightarrow
V_n^\ell\otimes_{\EOH_{n+1}^\ell} U_n^\ell
is an isomorphism,
where $\sigma_n$ is as in \cref{mate2} or the zero map
in the extremal case $n=0,n'=\ell-1$.
By \cref{lemma3},
the domain of $f$ is free as a right $\EOH_n^\ell$-supermodule
with basis $\big\{u_{n-1}(x^r) \otimes v_{n-1}(x^s)\:\big|\:
0 \leq r \leq n'+1, 0 \leq s \leq n-1\big\}
\cup \big\{b_1,\dots,b_{n'-n+1}\big\}$,
where $b_i$ is the identity element in the $i$th copy of $\EOH_n^\ell$,
and the codomain of $f$ is
free as a right $\EOH_n^\ell$-supermodule with basis
$\big\{v_n(x^s)\otimes u_n(x^r)\:\big|\:0 \leq r \leq n',
0 \leq s \leq n\big\}$.
Both of these sets are of size $nn'+n+n'+1$, so it suffices to
see that $f$ is surjective. Again, we apply $-\otimes_{\EOH_n^\ell} \k$
and show that the resulting map $\bar f := f\otimes 1$ is surjective.
Let $uv(r,s) := u_{n-1}(x^r)\otimes v_{n-1}(x^s) \otimes 1$
and $vu(s,r) := v_n(x^s)\otimes u_n(x^r) \otimes 1$ for short.
So the domain of $\bar f$ has linear basis
0 \leq r \leq n'+1, 0 \leq s \leq n-1\}
\cup \{b_1\otimes 1,\dots,b_{n'-n+1}\otimes 1\}$
and the codomain has linear basis
$\{vu(s,r)\:|\:0 \leq r \leq n',
0 \leq s \leq n\}$.
By \cref{Ian} and \cref{Jackdoor}, we have that
\begin{align}\label{minimap}
\bar f\big(uv(r,s)\big) &=
\pm vu(n,r+s-n)
\pm vu(s,r),
\bar f(b_i\otimes 1) &=
\end{align}
for $0 \leq r\leq n'+1$, $0 \leq s \leq n-1$
and $1 \leq i \leq n'-n+1$,
interpreting $vu(r+s-n)$ as zero if $r+s-n < 0$
and $s$.
The proof is completed by the following.
\begin{itemize}
\item
We get $vu(n,r)$ for $0 \leq r \leq n'-n$ from the
images of the vectors $b_i\otimes 1$ for $i=1,\dots,n'-n+1$.
\item
We get $vu(n,r)$ for $n'-n+1\leq r \leq n'$ from the images of the
vectors $uv(n'+1,r)$ for $0 \leq r \leq n-1$. This uses the observation
that $vu(r,n'+1) = 0$.
\item Modulo the span of vectors already obtained, we get the remaining $vu(s,r)$ for $0 \leq s \leq n-1$
and $0 \leq r \leq n'$ from the images of the vectors
$uv(r,s)$ for the same values of $r$ and $s$.
\end{itemize}
\end{proof}

In the next lemma, we give explicit descriptions of the images of the leftward cups and caps under the graded 2-superfunctor $\Psi_\ell$ from \cref{actiontheorem}, that is, the superbimodule homomorphisms
\begin{align}
\coev'_n := \Psi_\ell\left(
\begin{tikzpicture}[anchorbase,scale=.8]
\draw[-] (0.4,0.3) to[out=-90, in=0] (0.1,-0.2);
\draw[->] (0.1,-0.2) to[out = 180, in = -90] (-0.2,0.3);
\node at (0.99,-0.1) {$\red\scriptstyle{2n-\ell+2}$};
\end{tikzpicture}\right)&:
\EOH_{n+1}^\ell\rightarrow
Q^{2n-\ell + 1} U_n^\ell \otimes_{\EOH_n^\ell} V_n^\ell,
\\
\ev'_n :=
\Psi_\ell\left(
\begin{tikzpicture}[anchorbase,scale=.8]
\draw[-] (0.4,-0.2) to[out=90, in=0] (0.1,0.3);
\draw[->] (0.1,0.3) to[out = 180, in = 90] (-0.2,-0.2);
\node at (.8,0.2) {$\red\scriptstyle{2n-\ell}$};
\end{tikzpicture}
\right)&: Q^{2n-\ell+1} V_n^\ell
\otimes_{\EOH_{n+1}^\ell} U_{n}^\ell \rightarrow \EOH_n^\ell
\end{align}
for $0 \leq n \leq \ell-1$ (these maps are zero for all other $n$).
Let $n'$ be defined so that $\ell=n+1+n'$.
Then, by \cref{table},
is a superbimodule homomorphism of
degree $n'-n$ and parity $n'-n\pmod{2}$,
is of degree $n-n'$ and parity $n-n'\pmod{2}$.
Recall also the maps $\tev_n$ and $\tcoev_n$ from \cref{secondadjunction}.

\begin{lemma}\label{leftwardscupsandcaps}
For $\ell=n+1+n'$ as above, we have that
\begin{align*}
\coev_n' &= (-1)^{\binom{n}{2}+(n+1)n'}
(p_{n'}^{-1} \otimes q_n) \circ \tcoev_n,&
\ev_n' &= (-1)^{\binom{n+1}{2}+(n+1)n'} \tev_n \circ (q_n^{-1} \otimes p_{n'}),
\end{align*}
$p_{n'}:Q^{-n} U_n^\ell \rightarrow \tU_n^\ell$
and $q_n:\tV_n^\ell \rightarrow Q^{3n-\ell+1} V_n^\ell$
are the superbimodule isomorphisms that are the identity maps
on the underlying vector spaces.
\begin{align}\label{bubbly1}
\displaystyle\Psi_\ell\left(\begin{tikzpicture}[anchorbase,scale=.8]
\draw[<-] (0,0.4) to[out=180,in=90] (-.2,0.2);
\draw[-] (0.2,0.2) to[out=90,in=0] (0,.4);
\draw[-] (-.2,0.2) to[out=-90,in=180] (0,0);
\draw[-] (0,0) to[out=0,in=-90] (0.2,0.2);
\node at (0.85,0.2) {$\red\scriptstyle{n-n'+1}$};
\opendot{-0.2,0.2};
\node at (-0.9,0.24) {$\scriptstyle{n-n'+r}$};
\end{tikzpicture}\right)(1) &=
\sum_{s=0}^r
%With s and r-s switched: (-1)^{n(r+s)+n'r+\binom{r+s}{2}}
\Big[\big(\psi_{n+1}^\ell\big)^{-1}\big(\bar\eps_{r-s}^{(n')}\big) \Big]\bar\eta_{s}^{(n+1)}
%\bar\gamm_{r-s}^{(n+1)} \psi_{n'}^\ell\big(\bar\eps_s^{(n')}\big)
&&\text{if $r \geq n'-n$,}\\
\label{bubbly2}
\displaystyle\Psi_\ell\left(\begin{tikzpicture}[anchorbase,scale=.8]
\draw[->] (0.2,0.2) to[out=90,in=0] (0,.4);
\draw[-] (0,0.4) to[out=180,in=90] (-.2,0.2);
\draw[-] (-.2,0.2) to[out=-90,in=180] (0,0);
\draw[-] (0,0) to[out=0,in=-90] (0.2,0.2);
\node at (-0.85,0.2) {$\red\scriptstyle{n-n'-1}$};
\opendot{0.2,0.2};
\node at (0.9,0.22) {$\scriptstyle{n'-n+r}$};
\end{tikzpicture}\right)(1) &=
\sum_{s=0}^r (-1)^{(n'+s)r+(n+1)s+\binom{s}{2}}
\Big[\big(\psi_{n}^\ell\big)^{-1} \big(\bar\gamm_{r-s}^{(n'+1)}\big)\Big]\bar\eps_s^{(n)}&&\text{if $r \geq n-n'$.}
\end{align}
\end{lemma}

\begin{proof}
Recalling \cref{evil}, the map $p_{n'}$ is of degree $n-2n'$ and parity $n'\pmod{2}$,
and $q_n$ is of degree $n-\ell+1$ and parity $n\pmod{2}$.
The inverse of the map $p_{n'}^{-1} \otimes q_n$
is $(-1)^{nn'} p_{n'} \otimes q_n^{-1}$.
With this in mind,
we let \begin{align*}
\widehat{\coev}_n &:= (-1)^{\binom{n}{2}+n'}
\big(p_{n'} \otimes q_n^{-1}\big) \circ \coev_{n}',
\widehat{\ev}_n &:= (-1)^{\binom{n+1}{2}+n'} \ev_n'
\circ \big(q_n \otimes p_{n'}^{-1}\big).
\end{align*}
These are both even of degree 0.
To prove the first part of the lemma, we must show that
$\widehat{\coev}_n = \tcoev_n$
and $\widehat{\ev}_n = \tev_n$.

We first show that
$\widehat{\coev}_n$ and $\widehat{\ev}_n$
are the counit and unit of an adjunction.
This follows from the left adjunction relations \cref{leftadj}:
\begin{align*}
\big(\id\otimes \widehat{\ev}_n\big)
\circ \big(\widehat{\coev}_n \otimes \id\big)
(-1)^{\binom{n}{2}+\binom{n+1}{2}} (\id \otimes \ev_n') \circ \big(\id \otimes q_n \otimes p_{n'}^{-1}\big)
\circ \big(p_{n'}\otimes q_n^{-1}\otimes \id\big)
\circ (\coev_n' \otimes \id)\\
(-1)^{n+n'} (\id \otimes \ev_n') \circ \big(p_{n'} \otimes \id \otimes \id\big)
\circ \big(\id\otimes \id\otimes p_{n'}^{-1}\big)
\circ (\coev_n' \otimes \id)\\
(-1)^{\ell-1} p_{n'} \circ (\id \otimes \ev_n')
\circ (\coev_n' \otimes \id) \circ p_{n'}^{-1}
= \id,\\
\big(\widehat{\ev}_n\otimes\id\big)
\circ \big(\id\otimes \widehat{\coev}_n \big)
(-1)^{\binom{n}{2}+\binom{n+1}{2}} (\ev_n' \otimes \id) \circ \big(q_n \otimes p_{n'}^{-1}\otimes\id\big)
\circ \big(\id\otimes p_{n'}\otimes q_n^{-1}\big)
\circ (\id\otimes \coev_n')\\
( \ev_n'\otimes\id) \circ \big(\id\otimes\id\otimes q_{n}^{-1}\big)
\circ \big(q_n\otimes \id\otimes \id\big)
\circ (\id\otimes \coev_n')\\
q_{n}^{-1} \circ (\ev_n'\otimes\id)
\circ (\id\otimes \coev_n') \circ q_{n}
= \id.
\end{align*}
Here, for brevity, we are implicitly assuming that $\OGBim_\ell$
is strict as in \cref{howstrict}.

So now we have two adjunctions
making $(\tV^\ell_n, \tU_n^\ell)$
into a dual pair, one $A_1$ with unit $\widehat{\ev}_n$ and counit
$\widehat{\coev}_n$ just constructed,
and the other $A_2$ with unit $\tev_n$ and counit
$\tcoev_n$ coming from \cref{secondadjunction}.
Any such adjunction $A$ induces a  degree 0 even
$\alpha:\tU^\ell_n\stackrel{\sim}{\rightarrow}\Hom_{\EOH_{n}^\ell\dash}(\tV_n^\ell, \EOH_{n}^\ell)$.
So from $A_1$ and $A_2$
we get isomorphisms $\alpha_1$ and $\alpha_2$,
hence, an even degree 0 automorphism
$\alpha_2^{-1} \circ \alpha_1$ of $\tU^\ell_n$.
By \cref{lemma3}(2c), $\tU^\ell_n$ is cyclic generated by the vector
$\tu_n(1)$. Moreover, this vector spans the (one-dimensional) graded component of $\tU^\ell_n$ of lowest degree.
So we must have that
$\alpha_2^{-1} \circ \alpha_1 = c_n \id$
for $c_n \in \k^\times$.

The argument in the previous paragraph
shows that
$\widehat{\coev}_n = c_n\; \tcoev_n$
and $\widehat{\ev}_n = c_n^{-1}\; \tev_n$
for some $c_n \in \k^\times$.
To complete the proof of the first part of the lemma, we must show that $c_n = 1$. To see this, we will show that
\begin{align}
\displaystyle\Psi_\ell\left(\begin{tikzpicture}[anchorbase,scale=.8]
\draw[<-] (0,0.4) to[out=180,in=90] (-.2,0.2);
\draw[-] (0.2,0.2) to[out=90,in=0] (0,.4);
\draw[-] (-.2,0.2) to[out=-90,in=180] (0,0);
\draw[-] (0,0) to[out=0,in=-90] (0.2,0.2);
\node at (0.85,0.2) {$\red\scriptstyle{n-n'+1}$};
\opendot{-0.2,0.2};
\node at (-0.9,0.24) {$\scriptstyle{n-n'+r}$};
\end{tikzpicture}\right)(1) &= c_n
\sum_{s=0}^r
%With s and r-s switched: (-1)^{n(r+s)+n'r+\binom{r+s}{2}}
\Big[\big(\psi_{n+1}^\ell\big)^{-1}\big(\bar\eps_{r-s}^{(n')}\big) \big]\bar\eta_{s}^{(n+1)}
&&\text{if $r \geq n'-n$,}\label{firstofthelast}\\\label{secondofthelast}
\displaystyle\Psi_\ell\left(\begin{tikzpicture}[anchorbase,scale=.8]
\draw[->] (0.2,0.2) to[out=90,in=0] (0,.4);
\draw[-] (0,0.4) to[out=180,in=90] (-.2,0.2);
\draw[-] (-.2,0.2) to[out=-90,in=180] (0,0);
\draw[-] (0,0) to[out=0,in=-90] (0.2,0.2);
\node at (-0.85,0.2) {$\red\scriptstyle{n-n'-1}$};
\opendot{0.2,0.2};
\node at (0.9,0.22) {$\scriptstyle{n'-n+r}$};
\end{tikzpicture}\right)(1) &=
c_n^{-1} \sum_{s=0}^r(-1)^{(n'+s)r+(n+1)s+\binom{s}{2}}
\Big[\big(\psi_{n}^\ell\big)^{-1} \big(\bar\gamm_{r-s}^{(n'+1)}\big)\Big]\bar\eps_s^{(n)}
&&\text{if $r \geq n-n'$.}
\end{align}
Given this, taking $r=0$ in one of these equations and using that the
``bottom bubbles"
\draw[<-] (0,0.4) to[out=180,in=90] (-.2,0.2);
\draw[-] (0.2,0.2) to[out=90,in=0] (0,.4);
\draw[-] (-.2,0.2) to[out=-90,in=180] (0,0);
\draw[-] (0,0) to[out=0,in=-90] (0.2,0.2);
\node at (0.85,0.2) {$\red\scriptstyle{n-n'+1}$};
\opendot{-0.2,0.2};
\node at (-0.64,0.24) {$\scriptstyle{n-n'}$};
\end{tikzpicture}$
\begin{tikzpicture}[anchorbase,scale=.8]
\draw[->] (0.2,0.2) to[out=90,in=0] (0,.4);
\draw[-] (0,0.4) to[out=180,in=90] (-.2,0.2);
\draw[-] (-.2,0.2) to[out=-90,in=180] (0,0);
\draw[-] (0,0) to[out=0,in=-90] (0.2,0.2);
\node at (-0.85,0.2) {$\red\scriptstyle{n-n'-1}$};
\opendot{0.2,0.2};
\node at (0.66,0.22) {$\scriptstyle{n'-n}$};
\end{tikzpicture}
are identities if $n \geq n'$ or $n \leq n'$, respectively,
gives that $c_n=1$, and the lemma follows.

In this paragraph, we prove \cref{firstofthelast}.
We need to apply
$\ev_n \circ \big(\rho_{(1);n}(x)^{n-n'+r}\otimes\id\big)
\circ \coev_n'$ to $1 \in \EOH_{n+1}^\ell$
using that $\coev_n' = (-1)^{\binom{n}{2}+(n+1)n'}
c_n\;\big(p_{n'}^{-1} \otimes q_n\big) \circ \tcoev_{n}$.
Applying $\tcoev_n$ to 1 using the second formula for that in \cref{secondadjunction} gives
\sum_{s=0}^{n'}(-1)^{\ell s
+\binom{s}{2}}\Big[\big(\psi_{n+1}^\ell\big)^{-1} \big(\bar\eps_{n'-s}^{(n')}\big)\Big]\tu_n(1)\otimes \tv_n(x^s).
Then we scale by
$(-1)^{\binom{n}{2}+(n+1)n'} c_n$ and apply
$(p_{n'}^{-1} \otimes q_n)$ to get
c_n \sum_{s=0}^{n'}(-1)^{\binom{n}{2}+(n+1)n'+\ell s
\big(\bar\eps_{n'-s}^{(n')}\big)\Big] u_n(1)\otimes v_n(x^s).
This is $\coev'_n(1)$.
Then we apply $\rho_{(1);n}(x)^{n-n'+r}\otimes\id$
(the dots on the left boundary of the bubble)
using \cref{out}
to get
c_n \sum_{s=0}^{n'}(-1)^{\binom{n}{2}+(n+1)n'+\ell s
+\binom{s}{2}+ns+n'(n'-s)+(n-n'+r)(n'-s)+\binom{n-n'+r}{2}}\Big[\big(\psi_{n+1}^\ell\big)^{-1} \big(\bar\eps_{n'-s}^{(n')}\big)\Big]\tu_n(x^{n-n'+r})\otimes \tv_n(x^s).
Finally we apply $\ev_n$ using the formula from \cref{reallynot}
to obtain
c_n \sum_{s=n'-r}^{n'}
(-1)^{\binom{n}{2}+(n+1)n'+\ell s +
\binom{s}{2}+ns+n'(n'-s)+(n-n'+r)(n'-s)+\binom{n-n'+r}{2}}
\Big[\big(\psi_{n+1}^\ell\big)^{-1}
\big(\bar\eps_{n'-s}^{(n')}\big)\Big]
\bar\gamm^{(n+1)}_{r+s-n'}.
It remains to reindex the summation replacing $s$ by $s+(n'-r)$
and to simplify the signs to obtain \cref{firstofthelast}.

To prove \cref{secondofthelast},
first note that
\draw[->] (0.2,0.2) to[out=90,in=0] (0,.4);
\draw[-] (0,0.4) to[out=180,in=90] (-.2,0.2);
\draw[-] (-.2,0.2) to[out=-90,in=180] (0,0);
\draw[-] (0,0) to[out=0,in=-90] (0.2,0.2);
\node at (-0.85,0.2) {$\red\scriptstyle{n-n'-1}$};
\opendot{0.2,0.2};
\node at (0.9,0.22) {$\scriptstyle{n'-n+r}$};
\end{tikzpicture}=
\draw[->] (0.2,0.2) to[out=90,in=0] (0,.4);
\draw[-] (0,0.4) to[out=180,in=90] (-.2,0.2);
\draw[-] (-.2,0.2) to[out=-90,in=180] (0,0);
\draw[-] (0,0) to[out=0,in=-90] (0.2,0.2);
\node at (0.95,0.2) {$\red\scriptstyle{n-n'-1}$};
\opendot{-0.2,0.2};
\node at (-0.8,0.22) {$\scriptstyle{n'-n+r}$};
\end{tikzpicture}
by \cref{ds} and the super interchange law.
Also $\Psi_\ell\Big(\begin{tikzpicture}[anchorbase,scale=.8]
\draw[<-] (0.08,-.3) to (0.08,.4);
\opendot{0.08,0.05};
\node at (-.7,-.1) {$\red\scriptstyle{n-n-'1}$};
\end{tikzpicture}\Big)=\lambda_{n;(1)}(x_1)$
by the definition \cref{phew}, \cref{actiontheorem} and \cref{mate1}.
Now we calculate by applying
$(-1)^{\binom{n'-n+r}{2}}\ev'_n \circ \big(\lambda_{n;(1)}(x)^{n'-n+r}\otimes\id\big)
\circ \coev_n$ to $1 \in \EOH_{n}^\ell$
using that $\ev_n' = (-1)^{\binom{n+1}{2}+(n+1)n'}
c_n^{-1}\;\tev_n\circ \big(q_{n}^{-1} \otimes p_{n'}\big)$.
By \cref{Jackdoor}, we have that
\coev_n(1) =
\sum_{s=0}^n v_n\big(x^{s}\big) \otimes u_n(1) \bar \eps^{(n)}_{n-s}.
Then we scale by
c_n^{-1}$ and apply $\lambda_{n;(1)}(x)^{n'-n+r}\otimes\id$ to obtain
c_n^{-1} \sum_{s=0}^n (-1)^{\binom{n'-n+r}{2}+\binom{n+1}{2}+(n+1)n'}
v_n\big(x^{n'-n+r+s}\big)\otimes u_n(1) \bar \eps^{(n)}_{n-s}.
Next $q_n^{-1} \otimes p_{n'}$ gives
c_n^{-1} \sum_{s=0}^n (-1)^{\binom{n'-n+r}{2}+\binom{n+1}{2}+(n+1)n'+n'(n'+r+s)}
\tv_n\big(x^{n'-n+r+s}\big)\otimes \tu_n(1) \bar \eps^{(n)}_{n-s}.
Finally we apply $\tev_n$ using the formula in
\cref{secondadjunction} to obtain
c_n^{-1} \sum_{s=n-r}^n (-1)^{\binom{n'-n+r}{2}+\binom{n+1}{2}+(n+1)n'+n'(n'+r+s)+n(n'-n+r+s)+\binom{n'-n+r+s}{2}}
\Big[\big(\psi_{n}^\ell\big)^{-1} \Big(
\bar\gamm_{r+s-n}^{(n'+1)}\Big)\Big] \bar \eps^{(n)}_{n-s}.
It remains to replace $s$ by $n-s$ and simplify the sign to obtain \cref{secondofthelast}.
\end{proof}

The formulae for the positively
dotted bubbles in \cref{bubbly1,bubbly2}
are rather complicated.
To simplify, one can apply the homomorphism $\alpha_n^\ell$
from \cref{alphamap}, to obtain the following.

\begin{corollary}\label{counterclockwisebubbles}
For $\ell = n+n'$ and $k=n-n'$,
the graded superalgebra homomorphism defined by the composition
\begin{tikzcd}
\arrow[r,"\Psi_\ell"]& \End_{\EOH_n\dash\EOH_n}(\EOH_n)
\arrow[r,"\theta \mapsto \theta(1)"]&\EOH_n
\arrow[r,"\alpha_n^\ell"]&
\end{tikzcd}
is equal to the canonical quotient map $R \twoheadrightarrow R_{n'},
\dot c
\mapsto \dot c^{(n')}$.
\end{corollary}

\begin{proof}
We note first that this composition is indeed a graded superalgebra homomorphism. Now we use \cref{bubbly1,bubbly2} to show
for all $r \geq 1$ that it takes
$\dot \eps_r \mapsto \dot \eps_r^{(n')}$
in the case $k \geq 0$
and $\dot\eta_r \mapsto \dot \eta_r^{(n')}$
in the case $k \leq 0$.
The arguments are similar in the two cases, so we just give
the details for $k \geq 0$, i.e., $n \geq n'$.
If $\ell = 0$ the result is trivial, so we may assume $\ell > 0$,
hence, $n \geq 1$.
the definition of $\beta_k(\dot \eps_r)$ from \cref{hatbeta},
we apply \cref{bubbly1} with $n$ replaced by $n-1$ to get that
\Psi_\ell(\beta_k(\dot\eps_r))(1)
(-1)^{(n-n'+1)r}\sum_{s=0}^r (-1)^{(n'+1)r+ns+\binom{s}{2}}
\Big[\big(\psi_{n}^\ell\big)^{-1}\big(\bar\eps_{r-s}^{(n')}\big)\Big]\bar\gamm_{s}^{(n)}.
From \cref{po2}, it follows that $\alpha_n^\ell\Big(\big(\psi_{n}^\ell\big)^{-1}\big(\bar e_{r}^{(n')}\big)\Big) =
(-1)^{nr} \dot e_{r}^{(n')}$, hence,
$\alpha_n^\ell\Big(\big(\psi_{n}^\ell\big)^{-1}\big(\bar \eps_{r}^{(n')}\big)\Big) =
(-1)^{nr} \dot \eps_{r}^{(n')}$.
= (-1)^{(n-n'+1)r+(n'+1)r+nr}  \dot\eps_r^{(n')} = \dot \eps_r^{(n')}$.
\iffalse
By \cref{notsobad}, we have that
$\coev_n(1) =
\sum_{s=0}^n v_n\big(x^s\big)\otimes u_n(1) \bar\eps_{n-s}^{(n)}$.
Then we apply
$\id\otimes \rho_{(1);n}(x)^{r+n'-n}$
to obtain
\begin{align*}
\sum_{s=0}^n
v_n\big(x^s\big)\otimes u_n\big(x^{r+n'-n}\big) \bar\eps_{n-s}^{(n)}.
\end{align*}
Then we apply $\ev_n'$ using \cref{leftwardscupsandcaps} and \cref{secondadjunction}.
First we have to scale by $(-1)^{\binom{n+1}{2}+(n+1)n'}$
and apply $q_n^{-1}\otimes p_{n'}$ to get
\sum_{s=0}^n
\tv_n\big(x^s\big)\otimes \tu_n\big(x^{r+n'-n}\big) \bar\eps_{n-s}^{(n)}.
Then the application of $\tev_n$ gives
\sum_{s=n-r}^n
\psi_{n'+1}^\ell\big(\bar\gamm_{r+s-n}^{(n'+1)}\big) \bar \eps_{n-s}^{(n)}.
Some final simplifying,
replacing $\bar \eps_{n-s}^{(n)}$ with $(-1)^{\binom{n-s}{2}} \bar e_{n-s}^{(n)}$ and
remembering to replace $n'+1$ with $n'$ at the end, gives
\cref{Theresult}.
\fi
\end{proof}

The results so far in this section have an application to prove the
{\em non-degeneracy} of $\UU(\sl_2)$, which was conjectured in [18, 3].
This asserts that the 2-morphism spaces in $\UU(\sl_2)$ have the expected
graded dimensions. The result may be formulated as follows.
For any $k,\ell \in \Z$
and 1-morphisms $X, Y \in \Hom_{\UU(\sl_2)}(k,\ell)$ (i.e., words
consisting of $m$ letters $E$ and $n$ letters $F$
such that $\ell = k + 2m-2n$)
we view the 2-morphism
$\Hom_{\UU(\sl_2)}(X,Y)$ as a graded right $R$-supermodule
so that $\dot c \in R$
acts by horizontally composing on the right with $\beta_{k}(\dot c)$.

\begin{theorem}\label{nondegthm}
For $k, \ell \in \Z$
$X, Y \in \Hom_{\UU(\sl_2)}(k,\ell)$,
the 2-morphism space
$\Hom_{\UU(\sl_2)}(X, Y)$
is free as a graded right $R$-supermodule
with basis given by a set of representatives for
equivalence classes of decorated
reduced $(X,Y)$-matchings
in the sense defined in \cite[Sec.8]{BE2}.
In particular, $\beta_k:R \rightarrow \End_{\UU(\sl_2)}(1_k)$
is an isomorphism for all $k \in \Z$.
\end{theorem}

\begin{proof}
The ``easy'' step in the proof is to show that
$\Hom_{\UU(\sl_2)}(X, Y)$ is spanned as a right $R$-supermodule
by the 2-morphisms that are the
representatives for
equivalence classes of decorated
reduced $(X,Y)$-matchings. This is proved by exhibiting
an explicit straightening algorithm
going by induction on the number of crossings. See \cite[Th.~8.1]{BE2}, which simply cites \cite[Prop.~3.11]{KL3} as
the argument is the same as in the purely even setting, or [13] for a more systematic treatment.
Note the straightening algorithm requires all of the relations described above, including the alternating braid relation.

The ``hard" step is to establish the linear independence.
By a standard reduction, which is again the same as in the ordinary even setting
as in \cite[Rem.~3.16]{KL3}, it suffices to treat the case that
$X = Y = E^d$ for some $d \geq 0$.
In this case, the
decorated reduced $(X,Y)$-matchings consist of
$d$ strings oriented from bottom to top decorated with some dots
close to the top boundary.
We index them by pairs $(\kappa,w)$ for $\kappa \in \N^d$
and $w \in S_d$.
For such a pair, the corresponding 2-morphism $f(\kappa,w)$
has $\kappa_i$ dots at the top of the $i$th string,
with the strings below arranged so that they represent some reduced expression for
Consider some linear relation
f := \sum_{\kappa\in \N^d,w \in S_d}
f(\kappa,w) \beta_k(\dot c_{\kappa,w}) = 0
for $\dot c_{\kappa,w} \in R$.
Each $\dot c_{\kappa,w}$ is an $\k$-linear combination
of basis vectors
$\dot h_\lambda$ of $R$ for $\lambda$ in some finite set $P_\kappa$
of partitions.
Pick $0\leq n \leq \ell$ with
$k = 2n-\ell$ in such a way that
$n$ and $\ell-n$ are both very large
relative to $|\kappa|$ and $|\lambda|$ for all
$\lambda \in P_\kappa,\kappa\in \N^n$ with $\dot c_{\kappa,w} \neq 0$
for some $w \in S_n$.
Then we apply the 2-superfunctor $\Psi_\ell$ to $f$ to obtain
the relation
\Psi_\ell(f) = \sum_{\kappa\in \N^d,w \in S_d}
\Psi_\ell(f(\kappa,w)) \Psi_\ell(\beta_k(\dot c_{\kappa,w})) = 0
in $\End_{\EOH_{n+d}^\ell\dash\EOH_n^\ell}\big(U_{n+d-1}\otimes_{\EOH_{n+d-1}^\ell}\cdots\otimes_{\EOH_{n+1}^\ell} U_n^\ell\big)$.
Conjugating with the isomorphism $b_{(1)^d;n}$
from \cref{ukraine2},
we get from $\Psi_\ell(f)$ a superbimodule
endomorphism $\tilde f = 0 $ of
Using \cref{inhand1,inhand2}, it follows that
\tilde f = \sum_{\kappa\in\N^d,w \in S_d}
\pm x^\kappa \tau_w \otimes
\Psi_\ell(\beta_k(\dot c_{\kappa,w}))
for some signs, where this is being
viewed as an endomorphism of the free right $\EOH_n^\ell$-superbimodule
using the right action of $\ONH_d$ from \cref{lemma2}(2).
By the large choice of $n$ and $\ell$,
the endomorphisms defined by each $x^\kappa \tau_w$
are linearly independent;
cf. the proof of \cref{ONHbasis}.
We deduce that
$\Psi_\ell\big(\beta_k(\dot c_{\kappa,w})\big) = 0$
for all $\kappa$ and $w$.

It remains to show that
$\Psi_\ell\big(\beta_k(\dot c_{\kappa,w})\big) = 0$
implies that $\dot c_{\kappa,w} = 0$ for sufficiently large $n$ and $\ell$.
Assume that
$\Psi_\ell\big(\beta_k(\dot c_{\kappa,w})\big) = 0$.
that $\dot c_{\kappa,w}$ is an $\k$-linear combination
of $\dot h_\lambda$ for $\lambda$ with $|\lambda|$ small,
this follows on evaluating at $1 \in \EOH_n^\ell$ then applying
the homomorphism $\alpha_n^\ell:\EOH_n^\ell \rightarrow R_{\ell-n}$.
The point here is that by
\cref{counterclockwisebubbles}
we have that
\alpha_n^\ell\Big(\Psi_\ell\big(\beta_k(\dot h_\lambda)\big)(1)\Big)
= \dot h_\lambda^{(\ell-n)}.
These elements of $R_{\ell-n}$ are linearly independent
for small $\lambda$, so we can conclude that the coefficients
of all $\dot h_\lambda$ in $\dot c_{\kappa,w}$ are zero.
\end{proof}

The following corollary is well known; see also
\cite[Th.~11.7]{BE2} for the
explicit definition of the isomorphism.
We just note a different convention for $(q,\pi)$-integers
is used
in \cite[Sec.~9]{BE2} compared to \cref{qinteger}.
This accounts for the difference in the defining
relation \cite[(9.2)]{BE2} for $U_{q,\pi}(\sl_2)$
compared to the relation \cref{fridaylate} being used for it here.

\begin{corollary}\label{sgr}
The split Grothendieck ring $K_0\big(\gsKarunderline(\UU(\sl_2))\big)$
is isomorphic as
a $\Z[q,q^{-1}]^\pi$-algebra
to the integral form
$\mathbf{U}_{q,\pi}(\sl_2)$ of $U_{q,\pi}(\sl_2)$ defined at the end of \cref{leavingdepoe}.
Under the isomorphism,
the isomorphism classes of the 1-morphisms
$E 1_k$ and $F 1_k$ correspond to the elements
of $\mathbf{U}_{q,\pi}(\sl_2)$ denoted by the same notation.
\end{corollary}

\begin{proof}
See \cite[Th.~12.1]{BE2}, which explains how to deduce this from the non-degeneracy
given by \cref{nondegthm}.
\end{proof}

\begin{remark}
\cref{nondegthm} is not new---it was already been established in [13] by a completely different technique.
Also a version of \cref{sgr} already appeared in [18].
The proof of \cref{nondegthm}
given here is in the same
spirit as the proof of non-degeneracy of the ordinary $\sl_2$ 2-category given in \cite[Prop.~8.2]{Lauda} and the more general proof of non-degeneracy for $\sl_n$ given in [28].
\end{remark}

For $d \geq 1$ and $k \in \Z$,
there are graded superalgebra homomorphisms
\begin{align}
\label{fromonh}
\rho_d^{(k)}:\ONH_d &\rightarrow \End_{\UU(\sl_2)}(E^d1_k)^{\sop},&&\\\notag
x_i &\mapsto
\begin{tikzpicture}[anchorbase,scale=.8]
\draw[->] (1,-.3) to (1,.4);
\node at (.5,.05) {$\cdots$};
\node at (-.5,.05) {$\cdots$};
\draw[->] (-1,-.3) to (-1,.4);
\draw[->] (0.0,-.3) to (0.0,.4);
\node at (0,-.5) {$\stringnumber{i}$};
\node at (1,-.5) {$\stringnumber{1}$};
\node at (-1,-.5) {$\stringnumber{d}$};
\opendot{0.0,0.05};
\node at (1.2,.1) {$\red\scriptstyle{k}$};
\end{tikzpicture},
&\tau_j &\mapsto
\begin{tikzpicture}[anchorbase,scale=.8]
\draw[->] (1,-.3) to (1,.4);
\node at (.5,.05) {$\cdots$};
\node at (-.8,.05) {$\cdots$};
\draw[->] (-1.3,-.3) to (-1.3,.4);
\draw[->] (0.0,-.3) to (-0.3,.4);
\draw[->] (-.3,-.3) to (0.0,.4);
\node at (0.03,-.5) {$\stringnumber{j}$};
\node at (-0.35,-.5) {$\stringnumber{j+1}$};
\node at (1,-.5) {$\stringnumber{1}$};
\node at (-1.3,-.5) {$\stringnumber{d}$};
\node at (1.2,.1) {$\red\scriptstyle{k}$};
\end{tikzpicture}\\
\lambda_d^{(k)}:\ONH_d &\rightarrow \End_{\UU(\sl_2)}(1_k F^d),&&\label{fromonh2}\\\notag
x_i &\mapsto
\begin{tikzpicture}[anchorbase,scale=.8]
\draw[<-] (1,-.3) to (1,.4);
\node at (.5,.05) {$\cdots$};
\node at (-.5,.05) {$\cdots$};
\draw[<-] (-1,-.3) to (-1,.4);
\draw[<-] (0.0,-.3) to (0.0,.4);
\node at (0,.6) {$\stringnumber{i}$};
\node at (-1,.6) {$\stringnumber{1}$};
\node at (1,.6) {$\stringnumber{d}$};
\opendot{0.0,0.07};
\node at (-1.2,.1) {$\red\scriptstyle{k}$};
\end{tikzpicture},
&\tau_j &\mapsto
\begin{tikzpicture}[anchorbase,scale=.8]
\draw[<-] (1,-.3) to (1,.4);
\node at (.5,.05) {$\cdots$};
\node at (-.8,.05) {$\cdots$};
\draw[<-] (-1.3,-.3) to (-1.3,.4);
\draw[<-] (0.0,-.3) to (-0.3,.4);
\draw[<-] (-.3,-.3) to (0.0,.4);
\node at (0.06,.6) {$\stringnumber{j+1}$};
\node at (-0.38,.6) {$\stringnumber{j}$};
\node at (1,.6) {$\stringnumber{d}$};
\node at (-1.3,.6) {$\stringnumber{1}$};
\node at (-1.5,.1) {$\red\scriptstyle{k}$};
\end{tikzpicture}
\end{align}
This follows from the relations \cref{nearlydone,nearlydone2},
with the signs in \cref{fromonh,fromonh2}
accounting for the difference between these and our preferred relations for $\ONH_n$
from \cref{ONH1,ONH2,ONH3,ONH4,ONH5,ONH6}.
Another consequence of \cref{nondegthm} is that
both $\rho_{d}^{(k)}$ and $\lambda_d^{(k)}$
are {\em injective}.

\begin{remark}\label{fromonhrem}
On comparing with \cref{inhand1,inhand2}, it follows that
the composition of $\rho_d^{(2n-\ell)}$
with the homomorphism
$\End_{\UU(\sl_2)}(E^d 1_k)^{\sop}\rightarrow
\End_{\EOH_{n+d}^\ell\dash \EOH_n^\ell}\big(Q^{-nd-\binom{d}{2}} U_{n+d-1}^\ell\otimes_{\EOH_{n+d-1}^\ell}
\cdots\otimes_{\EOH_{n+1}^\ell} U_n^\ell\big)^\sop$
induced by the 2-superfunctor
$\Psi_\ell$ is equal to
the anti-homomorphism $\rho_{(1^d);n}$ from \cref{cam} (up to a degree
One can check similarly starting from \cref{mate1} that the composition
of $\lambda_d^{(2n-\ell)}$
with the homomorphism induced by $\Psi_\ell$
is equal to the homomorphism $\lambda_{n;(1^d)}$ from \cref{smith} (up
to degree shift).
\end{remark}

To conclude the section, we explain how to define divided
In $\GSKar(\UU(\sl_2))$,
there are 1-morphisms
\begin{align}\label{divpowerdef}
E^{(d)} 1_k
Q^{\binom{d}{2}}\Big( E^d 1_k,
\rho^{(k)}_d((\chi\omega)_d)\Big)&:
k \rightarrow k+2d,\\\label{divpowerdef2}
1_{k}F^{(d)} :=
\Big(
1_k F^d ,\lambda^{(k)}_d((\omega\chi)_d)\Big)
&:k+2d \rightarrow k.
\end{align}
By \cref{borisagain}
plus \cref{idempotentstar}, we have that
\begin{align}\label{divpowers}
E^d 1_k &\simeq
\bigoplus_{w \in S_d}
Q^{2\ell(w) - \binom{d}{2}}
\Pi^{\ell(w)}
E^{(d)} 1_k,& 1_k F^d &\simeq \bigoplus_{w \in S_d}
Q^{2\ell(w) - \binom{d}{2}} \Pi^{\ell(w)}1_k F^{(d)}.
\end{align}
In view of \cref{poincare}, it follows that $E^{(d)} 1_k$
and $F^{(d)} 1_k$ categorify the divided powers
\cref{snow}, i.e., the isomorphism classes of the former 1-morphisms
under the isomorphism from
\cref{sgr} give the latter elements of $\mathbf{U}_{q,\pi}(\sl_2)$.
Note $E^{(d)} 1_k$ and $1_k F^{(d)}$ are
obtained by upshifting
the bottom degree summands of $E^d 1_k$ and $1_k F^d$.
It is also useful to have available the following, which are
downshifts of the top degree summands:
\begin{align}\label{divpowerdeftop}
\overline{E}^{(d)} 1_k
Q^{-\binom{d}{2}}\Big( E^d 1_k,
\rho^{(k)}_d((\omega\chi)_d)\Big)&:
k \rightarrow k+2d,\\\label{divpowerdeftop2}
1_k \overline{F}^{(d)}:=
\Big(
1_k F^d ,\lambda^{(k)}_d((\chi\omega)_d)\Big)
&:k+2d \rightarrow k.
\end{align}
These categorify the elements $\overline{E}^{(d)} 1_k$
and $1_k \overline{F}^{(d)}$
of $\mathbf{U}_{q,\pi}(\sl_2)$ from \cref{snowtilde} since,
by \cref{borisagain}
plus \cref{idempotentstar} again, we have that
\begin{align}\label{divpowerstilde}
E^d 1_k &\simeq
\bigoplus_{w \in S_d}
\Pi^{\ell(w)}
\overline{E}^{(d)} 1_k,&
1_k F^d &\simeq
\bigoplus_{w \in S_d}
\Pi^{\ell(w)}
1_k \overline{F}^{(d)}.
\end{align}
Mirroring \cref{snowier}, we have that
\begin{align}\label{snowiest}
\overline{E}^{(d)} 1_k &\simeq \Pi^{\binom{d}{2}} E^{(d)} 1_k,
1_k \overline{F}^{(d)} &\simeq \Pi^{\binom{d}{2}} 1_k F^{(d)}.
\end{align}
This follows because the idempotents
$(\omega\chi)_d$ and $(\chi \omega)_d$ are conjugate
as discussed after \cref{newsnow}\footnote{It could also be deduced from \cref{divpowers,divpowerstilde}
using Krull-Schmidt, but we prefer the argument given since it constructs the isomorphism explicitly.}.

\begin{lemma}\label{haverightadjs}
In $\GSKar(\UU(\sl_2))$,
the 1-morphism
$Q^{-d(k+d)}\Pi^{\binom{d}{2}} 1_k F^{(d)}$ is right dual to $E^{(d)} 1_k$,
and the 1-morphism
$Q^{d(k+d)} \Pi^{d(k+d)+\binom{d}{2}} E^{(d)} 1_{k-2d}$ is
right dual to $F^{(d)} 1_k$.
\end{lemma}

\begin{proof}
We first show that
$Q^{-d(k+d)} 1_k F^{(d)}$ is right dual to $E^{(d)} 1_k$
in the $(Q,\Pi)$-envelope $\GSKar(\UU(\sl_2))$.
By \cref{rightadj},
$Q^{-k-1} 1_k F$ is right dual to $E 1_k$.
Hence, $Q^{-d(k+d)} Q^{-\binom{d}{2}}
1_k F^d$ is right dual to $Q^{\binom{d}{2}} E^d 1_k$.
By definition, $E^{(d)} 1_k$ is the summand of
$Q^{\binom{d}{2}} E^d 1_k$ defined by the idempotent
$Q^{\binom{d}{2}} \rho_d^{(k)}\big((\chi\omega)_d\big)$
and $Q^{-d(k+d)} 1_k \overline{F}^{(d)}$ is the summand of
$Q^{-d(k+d)} Q^{-\binom{d}{2}} 1_k F^d$ defined by the
the idempotent
$Q^{-d(k+d)} Q^{-\binom{d}{2}} \lambda_d^{(k)}
\big((\chi\omega)_d\big)$.
Now we observe using \cref{oddadjunction}(2),
\cref{phew} and \cref{idempotentstar}
that $Q^{-d(k+d)}Q^{-\binom{d}{2}}\lambda_d^{(k)}\big((\chi\omega)_d\big)$
is the right mate of $Q^{\binom{d}{2}} \rho_d^{(k)}\big((\chi\omega)_d\big)$.
Hence, we get that
$Q^{-d(k+d)} 1_k \overline{F}^{(d)}$ is right dual to
$E^{(d)} 1_k$. It remains to apply
\cref{snowiest} to pass from
$Q^{-d(k+d)} 1_k \overline{F}^{(d)}$
$Q^{-d(k+d)} \Pi^{\binom{d}{2}}1_k F^{(d)}$.

The proof that
$Q^{d(k+d)}\Pi^{d(k+d)+\binom{d}{2}} E^{(d)} 1_{k}$ is
right dual to $1_k F^{(d)}$ is similar using \cref{leftadj} instead of \cref{rightadj}.
By \cref{leftadj} and \cref{oddadjunction}(1),
$Q^{k+1}\Pi^{k+1} E 1_k$ is right dual to $1_k F$.
$Q^{d(k+dx)-\binom{d}{2}} \Pi^{d(k+d)} E^d 1_k$ is right dual to
$Q^{\binom{d}{2}} 1_k F^d$.
Since \cref{crossingcyclicity} is more complicated than
\cref{phew}, it is no longer true that
the idempotent
\rho_d^{(k)}\big((\chi\omega)_d\big)$
is {\em equal} to the right mate of the idempotent
but these two idempotents are conjugate via even degree 0 units.
This follows by the Krull-Schmidt theorem applied to the finite-dimensional algebra that is the even degree 0 component of $\ONH_d \otimes R$.
Hence, $Q^{d(k+d)} \Pi^{d(k+d)} \overline{E}^{(d)} 1_k$ is right dual to
$1_k F^{(d)}$.
It remains to appeal to \cref{snowiest} one more time.
\end{proof}

\section{Some graded 2-representation theory}\label{2Repsec}

In this section, we develop some 2-representation theory of
the $\sl_2$ 2-category  $\UU(\sl_2)$ from
\cref{km2cat}. We work throughout in the graded setting, but all the
definitions and results here have analogs with the $\Z$-grading forgotten.
The following is modelled on \cite[Def.~5.1.1]{Rou}.

\begin{definition}\label{2repdef}
By a {\em graded 2-representation} $\cV$ of $\UU(\sl_2)$, we mean
a strict graded 2-superfunctor
$\cV:\UU(\sl_2)\rightarrow \GSCAT$.
Decoding the definition, $\cV$ consists of the following data:
\begin{itemize}
\item a graded supercategory $\cV$ with a
given decomposition
into {\em weight subcategories}
$\cV = \coprod_{k \in \Z} \cV_k$ (or
$\cV = \bigoplus_{k \in \Z} \cV_k$ when $\cV$ is additive);
\item graded superfunctors $E:\cV \rightarrow \cV$
and $F:\cV \rightarrow \cV$
such that $E|_{\cV_k}:\cV_k\rightarrow \cV_{k+2}$
and $F|_{\cV_k}:\cV_k \rightarrow \cV_{k-2}$ for each $k \in \Z$;
\item  graded supernatural transformations
$x:E \Rightarrow E$ and $\tau:E^2 \Rightarrow E^2$
which are odd of degrees 2
and $-2$, respectively;
\item (inhomogeneous) graded supernatural transformations
$\eta:\Id \Rightarrow FE$ and $\eps:EF \Rightarrow \Id$
$\eta: \Id_{\cV_k} \Rightarrow FE|_{\cV_k}$
$\eps:E F|_{\cV_{k+2}} \Rightarrow \Id_{\cV_{k+2}}$
are even of degrees $k+1$
and $-k-1$, respectively.
\end{itemize}
Then there are the axioms:
\begin{itemize}
\item
the relations from \cref{nearlydone} hold:
$\tau\circ \tau = 0$, $(\tau E) \circ (E \tau) \circ (\tau E) = (E \tau) \circ (\tau E) \circ (E \tau )$ and
$(Ex)\circ \tau+(xE) \circ \tau = (xE)\circ \tau +\tau \circ (Ex) = E^2$;
\item
$\eta$ and $\eps$ satisfy the zig-zag relations:
$(F\eps) \circ (\eta F) = F$
and $(\eps E) \circ (E \eta) = E$ (equivalently,
they define units and counits of adjunctions
making $Q^{-k-1} F|_{\cV_{k+2}}$ into a
right adjoint to $E|_{\cV_k}$ for each $k \in \Z$);
\item
letting $\sigma := (FE\eps) \circ (F\tau F) \circ (\eta EF):EF\Rightarrow FE$
be the image of the rightward crossing under $\cV$,
the following inhomogeneous matrices of
supernatural transformations are isomorphisms:
\begin{align*}
\left(\:\sigma
\qquad
\eps
\qquad
\eps\circ (xF)
\qquad
\cdots\qquad
\eps \circ (x F)^{k-1}
\right)^T&:
E F|_{\cV_k}\Rightarrow
F E|_{\cV_k} \oplus \Id_{\cV_k}^{\oplus k}&&
\text{for $k \geq 0$}\\
\left(\:
\sigma\qquad
\eta
\qquad
(F x) \circ \eta
\qquad\cdots\qquad
(F x)^{-k-1} \circ \eta
\right)
&:E F|_{\cV_k} \oplus  \Id_{\cV_k}^{\oplus (-k)}
\Rightarrow
F E|_{\cV_k}&&\text{for $k \leq 0$}
\end{align*}
\end{itemize}
\end{definition}

There are natural notations of (full) {\em sub-2-representations} (which are called ``invariant ideals" in \cite[$\S$4.2]{BD}),
{\em quotient 2-representations}, and
{\em morphisms} of graded 2-representations.
The latter definition, which is the super analog of \cite[Def.~2.3]{Rou},
is equivalent to the following, which is similar to the formulation adopted in
\cite[Sec.~5.2.1]{CR}; the terminology being used is the same as in
\cite[Def.~4.6]{BD} (and actually goes back to Ben Webster).

\begin{definition}\label{sef}
Let $\cV$ and $\cW$ be two graded 2-representations of $\UU(\sl_2)$.
A {\em strongly equivariant graded superfunctor}
$\Omega:\cV\rightarrow \cW$
is a graded superfunctor such that $\Omega|_{\cV_k}:\cV_k \rightarrow
\cW_k$ for each $k \in \Z$,
plus a degree 0 even graded
supernatural isomorphism
$\zeta:E \Omega \stackrel{\sim}{\Rightarrow} \Omega E$,
such that the following holds
\begin{itemize}
\item
the supernatural transformation $(F \Omega \eps) \circ (F \zeta F) \circ
(\eta \Omega F):\Omega F \Rightarrow F \Omega$ is invertible;
\item
we have that $(\Omega x) \circ \zeta = \zeta \circ (x \Omega)$;
\item we have that $(\Omega \tau) \circ (\zeta E)\circ (E \zeta) = (\zeta E) \circ (E \zeta) \circ (\tau \Omega)$.
\end{itemize}
A {\em strongly equivariant graded superequivalence} is
a strongly equivariant graded superfunctor which is also a superequivalence of
\end{definition}

\begin{remark}
For strongly equivariant graded superequivalences, the first axiom in \cref{sef} actually
holds automatically; see \cite[Rem.~4.8]{BD} where this is explained (in the purely even setting).
Also in [1], the diagrammatic interpretation of these definitions
is discussed, which we still find helpful.
\end{remark}

\begin{remark}
There is an obvious way to make the composition
of two strongly
equivariant graded superfunctors into a strongly equivariant graded superfunctor in its own
Also the identity functor $\Id$ is strongly equivariant with
$\zeta := 1_{E}$.
So there is a category $\mathpzc{Rep}(\UU(\sl_2))$
consisting of graded
2-representations and strongly equivariant graded superfunctors.
\end{remark}

Usually, the graded supercategories $\cV_k$ in
a graded 2-representation $\cV$ will have some extra structure,
such as being additive or $\Pi$-complete.
We are mainly interested here in what we call
{\em graded Karoubian 2-representations}.
By definition, this means a graded 2-representation $\cV$ such that,
for each $k \in \Z$, the weight subcategory
$\cV_k$ is additive and $\Pi$-complete,
and the underlying ordinary category
$\underline{\cV}_k$ is idempotent complete.
Any graded 2-representation $\cV$ can be upgraded to a Karoubian graded
2-representation by passing to its graded super Karoubi envelope $\GSKar(\cV)$.

Given a graded Karoubian 2-representation
$\cV$, the underlying graded 2-superfunctor
from $\UU(\sl_2)$ to $\cV$
extends canonically to a graded
2-superfunctor from the
graded super Karoubi envelope $\GSKar(\UU(\sl_2))$ to $\cV$.
The direct sum over all $k \in \Z$ of the images
under this graded 2-superfunctor
of the 1-morphisms
$E^{(d)} 1_k$ and $F^{(d)} 1_k$
from \cref{divpowerdef,divpowerdef2}
give graded superfunctors
\begin{equation}
E^{(d)}, F^{(d)}:\cV \rightarrow \cV.
\end{equation}
By \cref{divpowers}, we have that
\begin{align}\label{divpowersapp}
E^d &\simeq
\bigoplus_{w \in S_d}
\Pi^{\ell(w)}
Q^{2\ell(w) - \binom{d}{2}}
F^d &\simeq
\bigoplus_{w \in S_d}
\Pi^{\ell(w)}
Q^{2\ell(w) - \binom{d}{2}}
\end{align}
\cref{haverightadjs} implies that
F^{(d)}|_{\cV_{k+2d}}$ is right adjoint to
and $Q^{d(k+d)}\Pi^{d(k+d)+\binom{d}{2}}E^{(d)}|_{\cV_{k-2d}}$
is right adjoint to $F^{(d)}|_{\cV_{k}}$,
units and counits of adjunction that are defined by images of 2-morphisms in $\GSKar(\UU(\sl_2))$.

A graded 2-representation $\cV$
is said to be {\em integrable} if $E$ and $F$ are locally nilpotent, i.e.,
for any $k \in \Z$ and any $M \in \cV_k$
there is some $n \geq 0$ such that $E^n M = F^n M = 0$.
Also, for $\ell \in \N$, a {\em lowest weight object of weight $-\ell$}
means an object $M \in \cV_{-\ell}$ such that $FM = 0$.

\begin{example}\label{aishaonthephone}
Suppose that $\ell \in \N$. By
\cref{actiontheorem}, there is an integrable
graded Karoubian 2-representation
\begin{equation}
\EOH^\ell\psmod
\bigoplus_{n=0}^\ell \EOH_n^\ell\psmod
\end{equation}
with the weight $k$ subcategory
$(\EOH^\ell\psmod)_k$ equal to
$\EOH_n^\ell\psmod$ if $k = 2n-\ell$
for $0 \leq n \leq \ell$,
or the trivial (zero) graded supercategory otherwise.
Other data is as follows.
\begin{itemize}
\item
The graded superfunctors $E$ and $F$
are $Q^{-n} U_n^\ell\otimes_{\EOH_n^\ell}-$
on the weight subcategory $\EOH_n^\ell\psmod$
and $Q^{3n+1-\ell} V_n^\ell\otimes_{\EOH_{n+1}^\ell}-$
on the weight subcategory $\EOH_{n+1}^\ell\psmod$,
assuming $0 \leq n < \ell$. On all other weight subcategories, $E$ and $F$ are zero.
\item
The graded supernatural transformations $x$  and $\tau$ are defined
by the supernatural transformations $\rho_{(1);n}(x_1) \otimes \id$
viewed as elements
$\sEnd\big(Q^{-n} U_n^\ell\otimes_{\EOH_n^\ell}-\big)_{2,\1}$ and
the supernatural transformations
$-\rho_{(1^2);n}(\tau_1) \otimes \id$ viewed as elements of
$\sEnd\big(Q^{-2n-1} U_{n+1}^\ell\otimes_{\EOH_{n+1}^\ell}
Q^{-n} U_n^\ell\otimes_{\EOH_n^\ell}-\big)_{-2,\1}$, respectively,
for all admissible $n$.
\item
The graded supernatural transformations $\eta$ and $\eps$ are given by the appropriate
counit and unit from
\cref{cupsandcaps}.
% (viewed as superbimodule homomorphisms between
%superbimodules that are shifted in degree but not in parity compared to before).
%This 2-representation is obviously integrable and it is generated
%by the lowest weight object $\EONH_0^\ell$ of weight $-\ell$.
\item
The homomorphisms induced by \cref{fromonh,fromonh2}
are equal to \cref{cam,smith} thanks to \cref{fromonhrem}.
\item
For $0 \leq n \leq n+d \leq \ell$,
we have that
\simeq Q^{-dn} U_{(d);n}^\ell \otimes_{\EOH_n^\ell}-$
and $F^{(d)}|_{\EOH_{n+d}^\ell\psmod}
\simeq Q^{-d(\ell-3n-2d+1)}V_{n;(d)}^\ell \otimes_{\EOH_{n+d}^\ell}-$;
cf. \cref{K0}.
\end{itemize}
We point out also by \cref{K0} that
is naturally identified with the $\mathbf{U}_{q,\pi}(\sl_2)$-module
$\mathbf{V}(-\ell)$, and $\EOH_0^\ell$ is a lowest weight object of weight $-\ell$.
\end{example}

Now we come to one of the key constructions introduced by
Rouquier in [35] in the purely even case, the construction of
{\em cyclotomic quotients}.
For any $\ell \in \Z$,
there is a graded 2-representation $\cR(\ell)$
\begin{equation}
\cR(\ell)_k := \mathpzc{Hom}_{\UU(\sl_2)}(\ell,k)
\end{equation}
for $k \in \Z$,
viewed as a graded 2-representation of the graded 2-supercategory
$\UU(\sl_2)$ in an obvious way.
For example, the graded superfunctor
$E|_{\cR(\ell)_k}:\cR(\ell)_k \rightarrow \cR(\ell)_{k+2}$
is defined by horizontally composing on the left
with the 1-morphism $E 1_k$, and the
supernatural transformation $x:E|_{\cR(\ell)_k}\Rightarrow E|_{\cR(\ell)_k}$
is induced by the 2-endomorphism
\draw[->] (0.08,-.3) to (0.08,.4);
\opendot{0.08,0.05};
\node at (.35,.1) {$\red\scriptstyle{k}$};
\end{tikzpicture} :E 1_k \Rightarrow E 1_k$.
The graded 2-representation $\cR(\ell)$
has the following universal property.

\begin{lemma}\label{Lop}
Given any graded 2-representation $\cV$ and any $M \in \cV_\ell$
there is a canonical strongly equivariant graded superfunctor
$\omega_M:\cR(\ell) \rightarrow \cV$ taking the
object $1_\ell$ of $\cR(\ell)_{\ell}$ to $M$.
\end{lemma}

We define the {\em universal
graded 2-representation of lowest weight $-\ell \in \Z$},
denoted $\cV(-\ell)$, to be the quotient 2-representation
$\cR(-\ell) / \cI$,
where $\cI$ here is the sub-2-representation of $\cR(-\ell)$
generated by $\Big\downarrow \:{\red \scriptstyle -\ell}$ (the identity endomorphism of the object $F 1_{-\ell}$).
We denote the lowest weight object of $\cV(-\ell)_{-\ell}$
arising from the object $1_{-\ell} \in \cR(-\ell)_{-\ell}$
by $\overline{1}_{-\ell}$, and call this the {\em canonical
lowest weight object}. It is a generating object for $\cV(-\ell)$.
The identity endomorphism of
$\overline{1}_{-\ell}$ is equal to the image of
the bottom bubble
\draw[<-] (0,0.4) to[out=180,in=90] (-.2,0.2);
\draw[-] (0.2,0.2) to[out=90,in=0] (0,.4);
\draw[-] (-.2,0.2) to[out=-90,in=180] (0,0);
\draw[-] (0,0) to[out=0,in=-90] (0.2,0.2);
\node at (0.4,0.2) {$\red\scriptstyle{-\ell}$};
\opendot{-0.2,0.2};
\node at (-.7,0.2) {$\scriptstyle{-\ell-1}$};
\end{tikzpicture}$, i.e., the image of $1 \in R$
under the homomorphism \cref{hatbeta}.
If $\ell < 0$, this bubble is not a fake bubble, so it
belongs to $\cI$.
This shows that  the graded supercategory
$\cV(-\ell)$ is trivial if $\ell < 0$.
Thus, $\cV(-\ell)$ is only interesting if $\ell \in \N$,
i.e., it is a dominant weight for $\sl_2$.
The following, the universal property of $\cV(-\ell)$,
follows immediately from \cref{Lop} and
the universal property of quotients.

\begin{lemma}\label{Lup}
Let $\cV$ be any graded 2-representation
of $\UU(\sl_2)$, $\ell \in \N$ and
$M \in\cV_{-\ell}$ be a lowest weight object.
The superfunctor $\omega_M:\cR(-\ell)\rightarrow \cV$
from \cref{Lop} induces a strongly equivariant graded superfunctor
$\Omega_M:\cV(-\ell)\rightarrow \cV$
taking $\overline{1}_{-\ell}$ to $M$.
\end{lemma}

There is a more sophisticated version of \cref{Lup}, which is analogous to
\cite[Prop.~5.6]{Rou}. To formulate this, we need one more preliminary lemma.

\begin{lemma}\label{endofid}
The homomorphism\footnote{In fact, $\beta_{-\ell}$ is
itself an isomorphism thanks
to \cref{nondegthm}, but this result is not needed here.}
$\beta_{-\ell}:R \rightarrow \End_{\cR(-\ell)}(1_{-\ell})$
from \cref{hatbeta} induces an isomorphism
R_\ell\stackrel{\sim}{\rightarrow} \End_{\cV(-\ell)}(\overline{1}_{-\ell})$.
\end{lemma}

\begin{proof}
The bubble
\draw[<-] (0,0.4) to[out=180,in=90] (-.2,0.2);
\draw[-] (0.2,0.2) to[out=90,in=0] (0,.4);
\draw[-] (-.2,0.2) to[out=-90,in=180] (0,0);
\draw[-] (0,0) to[out=0,in=-90] (0.2,0.2);
\node at (0.5,0.2) {$\red\scriptstyle{-\ell}$};
\opendot{-0.2,0.2};
\node at (-.7,0.2) {$\scriptstyle{r-\ell-1}$};
\end{tikzpicture}$ belongs to $\cI$ for $r > \ell$.
Up to a sign, the composition of $\beta_{-\ell}$ with the canonical
map $\End_{\cR(-\ell)}(1_{-\ell}) \twoheadrightarrow \End_{\cV(-\ell)}(\overline{1}_{-\ell})$
takes $\dot \eps_r \in R$ to the
the image of this bubble, which is zero.
We deduce
that this  homomorphism factors through the quotient $R_\ell$ of $R$
to induce
Moreover, $\bar\beta_{-\ell}$ is surjective
since $\beta_{-\ell}$ is surjective by the ``easy" part of \cref{nondegthm}.

To show that $\bar\beta_{-\ell}$ is also injective, we use the following
diagram of graded supercategories and superfunctors:
\begin{tikzcd}
\END_{\UU(\sl_2)}(-\ell)\arrow[d,"\operatorname{Ev}_{\overline{1}_{-\ell}}" left]
\arrow[r,"\Psi_\ell" above]&\EOH_0^\ell \bisMod \EOH_0^\ell\arrow[d,"-\otimes_{\EOH_0^\ell} \EOH_0^\ell" right]\\
\cV(-\ell)_{-\ell}\arrow[r,"\Omega_{\EOH_0^\ell}" below]&\EOH_0^\ell\sMod
\end{tikzcd}
Here, the top map comes from \cref{actiontheorem},
the left hand vertical superfunctor
is given by evaluating
on the object $\overline{1}_{-\ell}$,
and the right hand vertical superfunctor is given by
tensoring with the lowest weight object $\EOH_0^\ell$.
The way the bottom superfunctor
$\Omega_{\EOH_0^\ell}$ is defined in \cref{Lup}
ensures that this diagram commutes strictly.
It follows that the middle square
in the following diagram commutes:
\begin{tikzcd}
\arrow[d,"\operatorname{can}" left,twoheadrightarrow]
R\arrow[r,"\beta_{-\ell}" above]&\End_{\cR(-\ell)_{-\ell}}(1_{-\ell})
\arrow[d,"\operatorname{can}" right]
\arrow[r,"\Psi_\ell" above]&\End_{\EOH_0^\ell\dash\EOH_0^\ell}(\EOH_0^\ell)
\arrow[d,"\theta\mapsto \theta\otimes \id" right]\\
\arrow[r,twoheadrightarrow,"\bar\beta_{-\ell}" below]&\End_{\cV(-\ell)_{-\ell}}(\overline{1}_{-\ell})\arrow[r,"\Omega_{\EOH_0^\ell}" below]&\End_{\EOH_0^\ell\dash}(\EOH_0^\ell)\arrow[r,"\phi \mapsto \phi(1)" below]&\EOH_0^\ell
\arrow[r,"\sim" above,"\alpha_0^\ell" below]&R_\ell
\end{tikzcd}
\cref{counterclockwisebubbles}
shows that
the composition $R \rightarrow R_\ell$
around the northeast boundary of this diagram
is equal to the canonical quotient map.
Hence, the composition $R_\ell\rightarrow R_\ell$
of the three maps at the bottom of the diagram is the identity.
This implies that $\bar\beta_{-\ell}$ is injective.
\end{proof}

Any morphism space
$\Hom_{\cR(-\ell)}(X,Y)$ in $\cR(-\ell)$
can be viewed as a right $R$-supermodule
so that $\ocirc c \in R$ acts by horizontally composing on the right
with $\beta_{-\ell}(\ocirc c)$.
This induces a structure of right $R_\ell$-supermodule
on any morphism space $\Hom_{\cV(-\ell)}(X,Y)$; cf.
the first paragraph of the proof of \cref{endofid}.
Given a graded $R_\ell$-superalgebra $A$,
we let $\cV(-\ell)\otimes_{R_\ell} A$
be the graded supercategory with the same
objects as $\cV(-\ell)$ and
morphism spaces $\Hom_{\cV(-\ell)\otimes_{R_\ell} A}(X,Y)
:= \Hom_{\cV(-\ell)}(X,Y)\otimes_{R_\ell} A$.
This is naturally a graded 2-representation
of $\UU(\sl_2)$ in its own right.

\begin{theorem}\label{structthm}
Let $\cV$ be any graded 2-representation
of $\UU(\sl_2)$, $\ell \in \N$ and
$M \in\cV_{-\ell}$ be any lowest weight object.
The strongly equivariant graded
superfunctor $\Omega_M:\cV(-\ell)\rightarrow
\cV$ from \cref{Lup} extends to a {\em fully faithful}
strongly equivariant graded superfunctor
$\Omega_M\otimes \id:\cV(-\ell)\otimes_{R_\ell}
\End_{\cV}(M) \rightarrow \cV$.
\end{theorem}

\begin{proof}
Let $A:= \End_{\cV}(M)$ for short.
The graded superfunctor $\Omega_M$ extends to
$\Omega_M\otimes \id$ by the universal property of tensor product.
To see that the resulting graded superfunctor is fully faithful,
we must show that it defines an isomorphism
$\Hom_{\cV(-\ell)\otimes_{R_\ell} A}(X,Y)
\stackrel{\sim}{\rightarrow}
\Hom_{\cV}(X,Y)$ for objects of any weight subcategory of
$\cV(-\ell)\otimes_{R_\ell} A$.
This is clear if $X=Y = \overline{1}_{-\ell}$.
The result in general then follows by the (now standard) technique
explained in the proof of \cite[Lem.~5.4, Prop.~5.6]{Rou}.
\end{proof}

\begin{corollary}\label{babykid}
For $\ell \in \N$, let $\EOH^\ell\psmod$ be the graded Karoubian 2-representation from \cref{aishaonthephone},
and let $\GSKar(\cV(-\ell))$ be the graded super Karoubi envelope of
$\cV(-\ell)$, which is another graded
Karoubian 2-representation.
The strongly equivariant graded superfunctor
$\Omega_{\EOH_0^\ell}:\cV(-\ell) \rightarrow \EOH^\ell\psmod$
associated to the lowest weight object
induces a strongly equivariant graded superequivalence
$\Xi_\ell:\GSKar(\cV(-\ell)) \rightarrow \EOH^\ell\psmod$.
\end{corollary}

\begin{proof}
In view of \cref{endofid,structthm},
$\Omega_{\EOH_0^\ell}$ is fully faithful.
This extends by the universal property of the graded super Karoubi envelope
to give
a fully faithful strongly equivariant graded superfunctor
$\Xi_\ell:\GSKar(\cV(-\ell)) \rightarrow \EOH^\ell\psmod$.
To see that $\Xi_\ell$ is a graded superequivalence, it remains to check that it is dense.
This follows because
E^{(n)} \EOH_0^\ell \simeq
U_{(n);0}^\ell \otimes_{\EOH_0^\ell} \EOH_0^\ell
\simeq \EOH_n^\ell,
the last isomorphism following
$U_{(n);0}^\ell$ is free of rank 1 as a graded left $\EOH_n^\ell$-supermodule
by \cref{lemma01}(2).
\end{proof}

We record one more basic lemma, which is analogous to the first part of \cite[Lem.~5.2]{Rou}.

\begin{lemma}\label{sillylemma}
Let $\cV$ be an integrable Karoubian graded 2-representation of $\UU(\sl_2)$.
Let $N$ be an object of $\cV_k$ for some $k \in \Z$.
$\Hom_{\cV_k}(E^n M, N) = 0$ for all $\ell \in \N$,
$n \geq 0$ such that $k = 2n-\ell$
and all lowest weight objects $M \in \cV_{-\ell}$, then $N = 0$.
\end{lemma}

\begin{proof}
Suppose that $N \neq 0$.
By integrability, there exists $n \geq 0$ such that
$F^n N \neq 0$ and $F^{n+1} N = 0$.
This means that $M := F^n N$ is a non-zero lowest weight object
of $\cV_{-\ell}$ for  $\ell=k-2n \in \N$.
By assumption, we have that $\Hom_{\cV_k}(E^n M, N) = 0$.
Hence, by adjunction, $$
\End_{\cV_{-\ell}}(M)=
\Hom_{\cV_{-\ell}}(M, F^n N)
\simeq \Hom_{\cV_k}(E^n M, N)
= 0.
It follows that $1_{M} = 0$, so $M = 0$, which is a contradiction.
\end{proof}

\begin{remark}
There is more still to be done here. For example, Rouquier continues in \cite[Sec.~5.1.4]{Rou} to construct
a Jordan-H\"older series in an arbitrary integrable
Karoubian 2-representation, and this result assuredly
carries over to our setting.
There is also a good theory of
{\em locally finite Abelian} 2-representations of $\UU(\sl_2)$,
including an analog of \cite[Prop.~5.20]{CR} which implies that
the irreducible objects of such a 2-representation
can be given the structure of a crystal in the sense of Kashiwara.
It would be worthwhile to extend \cite[Th.~5.27]{CR} (which is
a special case of Rouquier's ``control by $K_0$'' from \cite[Th.~5.22]{Rou})
to this setting. This would pave the way to more applications involving representations of the supergroup $Q(n)$ and
the Lie superalgebra $\mathfrak{q}_n(\C)$.
In the ordinary case, an alternative approach by-passing control by $K_0$ was developed in [5], which we expect
should also have an interesting and non-trivial super analog.
Another direction we would like to investigate further is
to extend \cref{actiontheorem,nondegthm} from odd $\sl_2$ to
the super Kac-Moody 2-category
associated to
``odd $\mathfrak{so}_{2n+1}$",
thereby giving an odd analog of the 2-representation
of $\mathfrak{sl}_n$ constructed
in [28].
\end{remark}

\section{The odd analog of the Rickard complex}\label{ricksec}

Let $\cV$ be a graded 2-supercategory.
The notation $\Ch^b(\cV)$ denotes the graded supercategory of bounded cochain complexes
and chain maps in $\cV$;
differentials in a cochain complex are assumed to be even of degree 0
but we allow
chain maps whose components are inhomogeneous.
Also $K^b(\cV)$ is the homotopy category, which
is a graded supercategory
with the same objects as
$\Ch^b(\cV)$ and morphisms that are chain homotopy equivalence classes of chain maps;
chain homotopies are again required
to be even of degree 0.
If $\cV$ is an
integrable graded Karoubian 2-representation of $\UU(\sl_2)$
as in the previous section,
both $\Ch^b(\cV)$ and $K^b(\cV)$ are themselves integrable graded Karoubian 2-representations
of $\UU(\sl_2)$ in a natural way.

Fix $k \in \Z$.
The {\em odd Rickard complex} $\Theta_k$,
so-called because it is the odd analog of the complex in \cite[Sec.~6.2]{CR}
which was
introduced originally by Rickard in the context of symmetric groups,
is the following
cochain complex in
\left\{
\begin{array}{rl}
\cdots \rightarrow
Q^d E^{(k+d)} F^{(d)}1_{-k} \stackrel{\partial^{-d}}{\rightarrow}
Q^{d-1}E^{(k+d-1)} F^{(d-1)} 1_{-k}\rightarrow
\cdots \rightarrow E^{(k)} 1_{-k} \rightarrow 0\rightarrow\cdots&\text{if $k \geq 0$}\\
\cdots \rightarrow
Q^d E^{(k+d)} F^{(d)}1_{-k} \stackrel{\partial^{-d}}{\rightarrow}
Q^{d-1}E^{(k+d-1)} F^{(d-1)} 1_{-k}\rightarrow
\cdots \rightarrow Q^{-k}F^{(-k)} 1_{-k} \rightarrow 0 \rightarrow \cdots
&\text{if $k \leq 0$},
\end{array}\right.
where in both cases $E^{(k+d)} F^{(d)} 1_{-k}$
is in cohomological degree $-d$.
The differential
\partial^{-d}:
Q^d E^{(k+d)} F^{(d)}1_{-k} \rightarrow
Q^{d-1} E^{(k+d-1)} F^{(d-1)} 1_{-k}
$$ is
the composition first of the ``inclusion"
of $Q^d E^{(k+d)} F^{(d)} 1_{-k}
\rightarrow Q^{k+3d-2} E^{(k+d-1)} E F F^{(d-1)} 1_{-k}$
as a summand\footnote{The idempotent endomorphism defining
$Q^{k+3d-2} E^{(k+d-1)} E F F^{(d-1)} 1_{-k}$ as a summand of $Q^d E^{k+d} F^d 1_{-k}$
decomposes as the sum of two mutually orthogonal idempotents, one of which
is the idempotent defining $Q^{d-1} E^{(k+d)} F^{(d)} 1_{-k}$.}
of $E^{(k+d-1)} E F F^{(d-1)} 1_{-k}$,
$Q^{k+3d-2} E^{(k+d-1)} \eps F^{(d-1)}:
Q^{k+3d-2} E^{(k+d-1)} E F F^{(d-1)} 1_{-k}
\rightarrow
Q^{d-1} E^{(k+d-1)} F^{(d-1)} 1_{-k}$.
Note this is even of degree 0 as required.
The following checks that it is a cochain complex.

\begin{lemma}\label{kentucky}
We have that $\partial^{-d+1} \circ \partial^{-d} = 0$ for all $d$.
\end{lemma}

\begin{proof}
Ignoring gradings for brevity,
it suffices to show that
the composition
E^{(2)} F^{(2)}
\stackrel{\operatorname{inc}}{\longrightarrow}
E^2 F^2 1_{-k-2d+4}
\stackrel{E \eps F}{\longrightarrow} E 1_{-k-2d+4}
F \stackrel{\eps}{\longrightarrow} 1_{-k-2d+4}
is zero.
The identity endomorphism of
$E^{(2)} F^{(2)} 1_{-k-2d+4}$
is $$
\rho_2^{(-k-2d)}(x_1 \tau_1) \lambda_2^{(-k-2d)} (\tau_1 x_1)
= \big(\rho_2^{(-k-2d)}(\tau_1) \lambda_2^{(-k-2d)}(\tau_1)\big)
\circ \big(\rho_2^{(-k-2d)}(x_1) \lambda_2^{(-k-2d)}(x_1)\big).
The composition of this with
$\eps \circ (E \eps F)$ is zero:
\begin{tikzpicture}[anchorbase,scale=1]
\draw[->] (-.1,-.5) to (-.1,.2) [out=90,in=-90] to (.2,.7) to [out=90,in=-90] (.2,.8) to [out=90,in=180](.4,1) to [out=0,in=90] (.6,.8) to[out=-90,in=90] (.6,.3) to [out=-90,in=90] (.9,-.2) to [out=-90,in=90] (0.9,-.5);
\draw[->] (.2,-.5) to (.2,.2) [out=90,in=-90] to (-.1,.7) to [out=90,in=-90] (-.1,.8) to [out=90,in=180] (.4,1.3) to [out=0,in=90]
(.9,.8) to [out=-90,in=90] (.9,.3) to [out=-90,in=90] (.6,-.2) to [out=-90,in=90] (.6,-.5);
\opendot{.2,-.2};
\opendot{.6,-.3};
\node at (1.5,.45) {$\red\scriptstyle{-k-2d+4}$};
\end{tikzpicture}
\begin{tikzpicture}[anchorbase,scale=1]
\draw[->] (.2,-.5) to [out=90,in=-90] (.2,.8) to [out=90,in=180] (.4,1) to[out=0,in=90] (.6,.8) to [out=-90,in=90] (.9,.3) to[out=-90,in=90] (.6,-.2) to [out=-90,in=90] (.6,-.5);
\draw[->] (-.1,-.5) to[out=90,in=-90] (-.1,.8) to [out=90,in=180] (.4,1.3) to [out=0,in=90] (.9,.8) to[out=-90,in=90] (.6,.3) to [out=-90,in=90] (.9,-.2) to [out=-90,in=90] (0.9,-.5);
\node at (1.5,.45) {$\red\scriptstyle{-k-2d+4}$};
\opendot{.2,-.2};
\opendot{.6,-.3};
\end{tikzpicture} = 0.
\end{proof}

\begin{remark}
Note \cref{kentucky} plus \cref{actiontheorem} implies \cref{isdifferential}.
So the proof of that lemma was actually unnecessary
(as, by association,
was \cref{mate3}) but we included it to make \cref{derby}
independent of the subsequent material.
%(\cref{mate1} has never been used either, but it serves as a useful warm-up for \cref{mate2} which is crucial for the proof of the key \cref{actiontheorem}.)
\end{remark}

Suppose now that $\cV$ is an integrable graded Karoubian 2-representation
of $\UU(\sl_2)$.
Given any cochain complex $C \in \Ch^b(\cV_{-k})$,
we can apply the complex of graded superfunctors that is the image under
$\cV$ of the odd Rickard complex $\Theta_k$ to obtain a double complex.
The associated total complex is again bounded thanks to the integrability assumption.
This construction defines a graded superfunctor
$\Ch^b(\cV_{-k}) \rightarrow \Ch^b(\cV_k)$. Passing to the quotient
$K^b(\cV)$ of $\Ch^b(\cV)$,
we obtain from this a graded superfunctor
\begin{equation}
\cV(\Theta_k):K^b(\cV_{-k}) \rightarrow K^b(\cV_k).
\end{equation}

\begin{lemma}\label{nearly}
Let $\cV$ be the graded 2-representation $\EOH^\ell\psmod$ from
\cref{aishaonthephone}. The image of the odd Rickard complex
$\Theta_k$ under $\cV$ recovers the singular Rouqiuer complex
from \cref{src} shifted globally in degree by an application of $Q^{-nk}$.
\end{lemma}

\begin{proof}
This follows using the explicit identification of the
divided powers $E^{(d)}$ and $F^{(d)}$ as
endofunctors of $\cV$ explained in \cref{aishaonthephone}.
We just check that the degree shifts match correctly.
Let $n = \frac{\ell-k}{2}$ and $d$ be as in \cref{weekend}, so $k = \ell-2n$.
In the $-d$th cohomological degree in the odd Rickard complex,
we have
$Q^d E^{(k+d)} F^{(d)}1_{-k}$.
In the 2-representation $\cV$, this acts by tensoring with the graded
\big(Q^{-(\ell-2n+d)(n-d)}
U_{(k+d);n-d}^\ell \big)\otimes_{\EOH_{n-d}^\ell} \big(Q^{-d(\ell-3(n-d)-2d+1)}
V_{n-d;(d)}^\ell\big)$, where
the degree shifts are as described in \cref{aishaonthephone}.
The total grading shift here simplifies to
$Q^{-n k}$,
so this is equal to the graded superbimodule
$U_{(k+d);n-d} \otimes_{\EOH_{n-d}^\ell} V_{n-d;(d)}$
in the $d$th homological degree of the singular Rouquier complex
shifted by $Q^{-nk}$.
\end{proof}

\begin{corollary}\label{dearly}
For $\ell \in \N$,
$\big(\GSKar(\cV(-\ell)\big))(\Theta_k):K^b\big(\GSKar(\cV(-\ell))_{-k}\big) \rightarrow K^b\big(\GSKar(\cV(-\ell))_k\big)$ is a graded superequivalence
inducing $T:1_{-k} \mathbf{V}(-\ell)\stackrel{\sim}{\rightarrow}
1_k \mathbf{V}(-\ell)$ at the level of the Grothendieck groups.
\end{corollary}

\begin{proof}
This follows from \cref{nearly} together with \cref{weirdshift} and \cref{babykid}.
\end{proof}

The proof of the following theorem is based on the argument in \cite[Th.~5.18]{Rou},
the main step really being \cite[Lem.~5.5]{Rou}.
This was itself a generalization of \cite[Th.~6.4]{CR}
constructed equivalences between
bounded derived categories of locally finite Abelian 2-representations.

\begin{theorem}\label{bigt}
Let $\cV$ be an integrable graded Karoubian 2-representation
of $\UU(\sl_2)$.
For $k \in \Z$, the graded
superfunctor $\cV(\Theta_k):K^b(\cV_{-k}) \rightarrow K^b(\cV_k)$ induced by the odd Rickard complex
is a graded superequivalence.
\end{theorem}

\begin{proof}
By \cref{haverightadjs}, the 1-morphism $Q^d E^{(k+d)} F^{(d)} 1_{-k}$ has a right dual
in $\GSKar(\UU(\sl_2))$.
Hence, we can form the right dual
$\Theta^k$ to $\Theta_k$, which is a cochain complex in
The 1-morphism in
the $d$th cohomological degree of $\Theta^k$
is the right dual
of the 1-morphism
in the $(-d)$th cohomological degree of $\Theta_k$,
and the differentials in $\Theta^k$ are the right mates
of the corresponding differentials in $\Theta_k$.
$\Theta^k \circ \Theta_k$ and $\Theta_k \circ \Theta^k$
be the total complexes associated to the double complexes obtained by
composing these cochain complexes.
The complex $\Theta_k$ is bounded above, and $\Theta^k$ is bounded below,
but neither is bounded. Consequently,
in each cohomological degree,
the total complexes
$\Theta^k \circ \Theta_k$ and $\Theta_k \circ \Theta^k$
involve {\em infinite} direct sums of 1-morphisms in
so in fact, one needs to pass to a completion of this graded $(Q,\Pi)$-supercategory
for it to make sense. This does not cause issues since, on a given object in an
integrable graded Karoubian
2-representation, the superfunctors arising from
all but finitely many of the summands of these infinite
direct sums are zero.

Like $\Theta_k$, the complex $\Theta^k$ defines a graded superfunctor
denoted $\cV(\Theta^k):K^b(\cV_k) \rightarrow
K^b(\cV_{-k})$. Moreover, $\cV(\Theta^k)$ is
right adjoint to $\cV(\Theta_k)$, with
counit and unit of adjunction denoted
\begin{align*}
\cV(\eps):&
\cV(\Theta_k)\circ\cV(\Theta^k)
\Rightarrow \Id_{K^b(\cV_k)},
\Rightarrow
\cV(\Theta^k) \circ \cV(\Theta_k).
\end{align*}
This is explained in more detail in \cite[Sec.~4.1.4]{CR}.
As the notation $\cV(\eps)$ and
$\cV(\eta)$ suggests, if we identify
$\cV(\Theta_k)\circ\cV(\Theta^k)$ with $\cV(\Theta_k \circ \Theta^k)$
$\cV(\Theta^k) \circ \cV(\Theta_k)$ with
$\cV(\Theta^k \circ \Theta_k)$ then
these even degree 0 supernatural transformations are induced by
corresponding chain maps denoted simply by
$\eps:\Theta_k\circ\Theta^k\Rightarrow 1_k$
and $\eta:1_{-k}\rightarrow \Theta^k\circ \Theta_k$
between cochain complexes in the completion of $\GSKar(\UU(\sl_2))$.
Although not needed here, these chain maps can be seen quite explicitly; the
matrix coefficients of their components
are 2-morphisms in $\GSKar(\UU(\sl_2))$ that arise from the counits
and units defining the duality between
the 1-morphisms $Q^d E^{(k+d)} F^{(d)} 1_{-k}$ and their right duals.

To prove the theorem, it suffices to show that
$\cV(\eps)$ and $\cV(\eta)$ are isomorphisms.
We just explain the argument to see this in the case of
$\cV(\eps)$, since the case of $\cV(\eta)$ is similar.
The even degree 0 graded supernatural transformation $\cV(\eps)$ is an isomorphism if and only if
$\Cone(\cV(\eps)_C) = 0$ for all $C \in K^b(\cV_k)$.
Now we observe that
$\Cone(\cV(\eps)_C) = \cV(Z)(C)$
where $Z := \Cone(\eps)$ is the cone of
$\eps:\Theta_k\circ\Theta^k\Rightarrow 1_k$.
%, that is, it
%is a cochain complex in the completion of $\GSKar(\UU(\sl_2))$
%equal to $\Theta_k\circ\Theta^k[1] \oplus 1_k$
%with the appropriate differential;
% $1_k$ here is being viewed as a cochain complex concentrated in cohomological
% degree 0.
Thus, it suffices to show that
the graded superfunctor $\cV(Z):K^b(\cV_k)\rightarrow K^b(\cV_k)$
is zero.

\iffalse
% OLD VERSION WHICH WASNT QUITE RIGHT!
Consider $K^b(\cV)$ as an integrable Karoubian graded 2-representation in its own right.
In this paragraph, we show that
\cV(Z)(E^n C) = 0
in $K^b(\cV_k)$
for all $\ell \in \N$, $n \geq 0$
such that $k=2n-\ell$,
and all lowest weight objects $C \in K^b(\cV_{-\ell})$.
To see this, we apply \cref{Lup} (with $\cV$ replaced by
to get a
strongly equivariant graded superfunctor $\Omega_C:\cV(-\ell)\rightarrow K^b(\cV)$
taking $\bar 1_{-\ell}$ to $C$.
This extends to a strongly equivariant graded superfunctor
$\widehat{\Omega}_C:\GSKar(\cV(-\ell)) \rightarrow K^b(\cV)$
by the universal property of
graded super Karoubi envelope.
The  graded superfunctor underlying $\widehat{\Omega}_C$
extends to
graded superfunctor
$\Ch^b(\GSKar(\cV(-\ell)_k)) \rightarrow \Ch^b(K^b(\cV_k))$,
K^b(\GSKar(\cV(-\ell)_k)) \rightarrow K^b(K^b(\cV_k))$.
Since $\widehat{\Omega}_C$ is strongly equivariant,
we have that
\Xi \circ (\GSKar(\cV(-\ell)))(Z) \simeq
K^b(\cV)(Z) \circ \Xi.
By \cref{dearly}, we know that
$(\GSKar(\cV(-\ell)))(Z) = 0$
in $K^b(\GSKar(\cV(-\ell)_k))$.
$\big(K^b(\cV)(Z)\big)(E^n C) = 0$ in $K^b(K^b(\cV_k))$
since we have that
$\Xi (E^n \bar 1_{-\ell}) \simeq E^n C$.
Viewing $K^b(\cV)(Z)(E^n C)$ as a double complex, this implies that the
corresponding total complex in $K^b(\cV_k)$ is zero, which
gives that $\cV(Z)(E^n C) = 0$ as required.
\fi

Consider $K^b(\cV)$ as an integrable Karoubian graded 2-representation in its own right.
In this paragraph, we show that
$\cV(Z)(E^n C) = 0$
in $K^b(\cV_k)$
for all $\ell \in \N$, $n \geq 0$
such that $k=2n-\ell$,
and all lowest weight objects $C \in K^b(\cV_{-\ell})$.
To see this, we apply \cref{Lup} (with $\cV$ replaced by
to get a
strongly equivariant graded superfunctor $\Omega_C:\cV(-\ell)\rightarrow K^b(\cV)$
taking $\bar 1_{-\ell}$ to $C$.
This extends to a strongly equivariant graded superfunctor
$\widehat{\Omega}_C:\GSKar(\cV(-\ell)) \rightarrow K^b(\cV)$
by the universal property of
graded super Karoubi envelope.
Let $K^b(\widehat{\Omega}_C):
K^b(\GSKar(\cV(-\ell)_k)) \rightarrow K^b(\cV_k)$
be the graded superfunctor defined by applying
$\widehat{\Omega}_C$ to a complex in
$K^b(\GSKar(\cV(-\ell)_k))$ to obtain a double complex then taking the associated total complex.
Since $\widehat{\Omega}_C$ is strongly equivariant,
we have that
\begin{equation}\label{lastthing}
\circ (\GSKar(\cV(-\ell)))(Z)\circ \operatorname{inc} \simeq
\cV(Z) \circ \widehat{\Omega}_C,
\end{equation}
where $\operatorname{inc}:\GSKar(\cV(-\ell))
\rightarrow K^b(\GSKar(\cV(-\ell))$ is the canonical graded superfunctor
sending objects to complexes concentrated in cohomological degree 0.
By \cref{dearly},
$(\GSKar(\cV(-\ell)))(Z) = 0$
in $K^b(\GSKar(\cV(-\ell)_k))$,
hence, the graded superfunctor on the left hand side of \cref{lastthing}
takes $E^n \bar 1_{-\ell}$ to 0.
So the graded superfunctor on the right hand side takes
$E^n \bar 1_{-\ell}$ to 0 too.
Since we have that
$\widehat{\Omega}_C (E^n \bar 1_{-\ell}) \simeq E^n C$,
it follows that
$\cV(Z)(E^n C) = 0$ as required.

To complete the proof,
we let $\cV(Z)^\vee$ be a right adjoint to
$\cV(Z):K^b(\cV_k) \rightarrow K^b(\cV_k)$, which exists by the general
discussion in \cite[Sec.~4.1.4]{CR} again.
We must show that $\cV(Z)(D) = 0$ for any
$D \in K^b(\cV_k)$, which we do by showing that
$\cV(Z)^\vee \big(\cV(Z) (D)\big) = 0$;
this is sufficient since
it implies that $\End_{K^b(\cV_k)}\big(\cV(Z)(D)\big) = 0$.
Using \cref{sillylemma}, we just need to show that
\Hom_{K^b(\cV_k)}\Big(E^n C,
\cV(Z)^\vee \big(\cV(Z) (D)\big)\Big) = 0
for $C$ and $n$ as in the previous paragraph.
This follows because by adjunction we have that
\Hom_{K^b(\cV_k)}\Big(E^n C,
\cV(Z)^\vee \big( \cV(Z) (D)\big)\Big)
\simeq
\Hom_{K^b(\cV_k)}\big(\cV(Z)(E^n C),
\cV(Z)(D)\big)
which is zero by the previous paragraph.
\end{proof}

\section{Application to representations of spin symmetric groups}\label{sapps}

\cref{bigt} can be applied to obtain graded superequivalences
between homotopy/derived categories of
supermodules over the
cyclotomic quiver Hecke superalgebras from [26, 23, 24].
In explaining this, we will mainly cite \cite[Sec.~8]{KKO2} which presents the results needed to do this
rather concisely. However,
we need to
reverse the roles of $E$ and $F$
compared to [24] to be consistent with our convention for $\UU(\sl_2)$
in \cref{catsec}, in which we preferred lowest weight modules
to highest weight modules.

Fix a Cartan superdatum $(A, P, \Pi, \Pi^\vee)$ as in \cite[Sec.~4.1]{KKO2}. So:
\begin{itemize}
\item
$I$ is an index set
with given decomposition $I = I_{\even}\sqcup I_{\odd}$;
\item
$A = (a_{i,j})_{i,j \in I}$ is a symmetrizable
Cartan matrix such that $a_{i,j} \in 2\Z$ for all $i \in I_{\odd}, j \in I$;
\item
$P$ is the weight lattice;
\item
$\Pi = \{\alpha_i\:|\:i \in I\}$ is the set of simple roots;
\item $\Pi^\vee = \{h_i\:|\:i \in I\}$ is the set of simple coroots.
\end{itemize}
Let $d_i\:(i \in I)$ be positive integers chosen
so that $d_i a_{i,j} = d_j a_{j,i}$ for all $i,j \in I$.
Let $P^+$ be the corresponding set of dominant weights
and $Q^+ := \bigoplus_{i \in I} \N \alpha_i$ be the non-negative
part of the root lattice.
Finally, let $W < \Aut(P)$ be the Weyl group.

Let $\Bbbk = \bigoplus_{d \geq 0} \Bbbk_d$ be a positively graded commutative ground ring with $\Bbbk_0 = \k$ (our usual algebraically closed ground field) and $\dim_\k
\Bbbk_d < \infty$ for all $d$. We view $\Bbbk$ as a purely even
graded $\k$-superalgebra.
Given any $\alpha \in Q^+$, there is a corresponding {\em quiver Hecke superalgebra} $R_\alpha$ which is defined by generators and relations
as in \cite[Sec.~8.1]{KKO2}; the definition depends on an
additional choice of parameters as explained in [24].
Let $R^\lambda_\alpha$ be the deformed cyclotomic quotient
from \cite[Def.~8.10]{KKO2} associated to a dominant weight
$\lambda \in P^+$ and a choice of
monic polynomials $a_i^\lambda \:(i \in I)$ as in \cite[(8.12)]{KKO2}.
We are interested in the graded $(Q,\Pi)$-supercategory
\begin{equation}
R^\lambda\psmod := \bigoplus_{\alpha \in Q^+}
\end{equation}
The constructions in \cite[Sec.~8.3]{KKO2} make $R^\lambda\psmod$
into a ``supercategorification" of the
integrable lowest weight module $V(-\lambda)$ for
the covering quantum group $U_{q,\pi}(\mathfrak{g})$
with the given Cartan superdatum.
From this, it can be seen that
$R^\lambda\psmod$ has the structure of a graded
2-representation of the corresponding graded Kac-Moody 2-supercategory
as defined in [3], with the Grothendieck group
being identified with the Kostant
$\Z[q,q^{-1}]^\pi$-form for

To be more precise, we focus now on some fixed $i \in I$
and consider the corresponding $\sl_2$-subalgebra
of $U_{q,\pi}(\mathfrak{g})$. In this generality,
we actually need to work now with $q_i := q^{d_i}$ and the grading shift functor
$Q_i := Q^{d_i}$ rather than $q$ and $Q$ used in previous sections.
This means that when $d_i > 1$
definitions such as \cref{2repdef}
earlier in the paper
should be modified by replacing $Q$ with $Q_i$
and scaling all degrees by $d_i$ too, e.g., $x$ and $\tau$
are now of degrees $2 d_i$ and $-2 d_i$ rather than of degrees 2 and $-2$.
Since the $\Z$- and $\Z/2$-gradings are independent this does not cause any problems.
There are graded superfunctors
\begin{align*}
R^\lambda \psmod \rightarrow R^\lambda \psmod,
F_i&:R^\lambda \psmod \rightarrow R^\lambda \psmod.
\end{align*}
In terms of the induction and restriction functors denoted
$F_i^\lambda$  and $E_i^\lambda$ in \cite[Sec.~8.3]{KKO2},
our $E_i$ is
$F_i^\lambda = \bigoplus_{\alpha \in Q_+} F_i^\lambda|_{R^\lambda_\alpha\psmod}$
and our $F_i$ is
$\bigoplus_{\alpha \in Q^+}
Q_i^{\langle h_i, \alpha-\lambda\rangle - 1} E_i^\lambda|_{R^\lambda_{\alpha}\psmod}$.
As well as switching the roles of $E$ and $F$
we have incorporated an additional grading shift
into the restriction functors compared to [24].
This is needed because [24] does not follow the standard
conventions for covering quantum groups.
It ensures that the graded supernatural transformations
$\eps:E_i F_i|_{R^\lambda_{\alpha+\alpha_i}\psmod} \Rightarrow \Id_{R^\lambda_{\alpha+\alpha_i}\psmod}$,
$\eta:\Id_{R^\lambda_\alpha\psmod} \Rightarrow F_i E_i|_{R^\lambda_\alpha\psmod}$
defined on a graded supermodule by exactly the same underlying functions as for the natural adjunction between restriction and induction
are of the correct degree to match the degrees of the rightward cups and caps
in \cref{table} (also now scaled by $d_i$).
Also in \cite[Sec.~8.3]{KKO2}, one finds the definition of graded supernatural transformations
$x:E_i \Rightarrow E_i$ of degree $2d_i$ and
$\tau:E_i^2 \Rightarrow E_i^2$
of degree $-2d_i$, both of which are even if $i \in I_{\even}$
and odd if $i \in I_{\odd}$.
(A further complication is that the language of supercategory, superfunctor and supernatural transformation is used differently in [24] compared to
here, but the appropriate translation is easy to make; see the table at the end of the introduction in [2].)

This construction makes $R^\lambda\psmod$ into
a graded integrable Karoubian 2-representation of the ordinary $\sl_2$
2-category from [30, 35]
if $i$ is even, or of our reduced odd $\sl_2$ 2-category
$\UU(\sl_2)$ as in \cref{2repdef}
if $i$ is odd (with the modified convention for degrees when $d_i >1$).
The last statement is not stated explicitly in [24]---the relevant place
is \cite[Th.~8.13]{KKO2} but one has to work through the proof which
goes back to \cite[Th.~5.2]{KK} to see that the isomorphisms are given by the
appropriate matrices of supernatural transformations needed to check
the difficult relations \cref{inv1,inv2}.
In the odd case,
the fact that the odd bubbles act as zero (as required by the final axiom in
\cref{2repdef}) follows
because they are zero on the generating lowest weight subcategory
$R^{\lambda}_0\psmod$ as that is purely even.

The following theorem now follows from \cite[Th.~5.18]{Rou}
if $i$ is even, with the graded superequivalence being induced by the
even analog of the Rickard complex,
or our \cref{bigt} if $i$ is odd.

\begin{theorem}
In the above setup, for $\alpha \in Q^+$ such that $V(-\lambda)_{\alpha-\lambda} \neq 0$, the even or odd Rickard complex $\Theta_{\langle h_i, \lambda-\alpha\rangle}$ induces a graded superequivalence
\rightarrow K^b\big(R^\lambda_{\alpha-\langle h_i, \alpha-\lambda\rangle \alpha_i}\psmod\big)$
categorifying the action of the $i$-th generator of the braid group
of $W$ on $V(-\lambda)$.
\end{theorem}

There is also a dual version of this theorem
with $R^\lambda\psmod$ replaced with
\begin{equation}
R^\lambda\smod := \bigoplus_{\alpha \in Q^+}
\end{equation}
where for a graded superalgebra $A$ we write
$A\smod$ for the graded supercategory of
graded left $A$-supermodules that are finite-dimensional over the
ground field $\k$.
The underlying ordinary category
is a locally finite Abelian $(Q,\Pi)$-category.
The results from \cite[Sec.~8.3]{KKO2}
show that this categorifies
the dual Kostant $\Z[q,q^{-1}]^\pi$-form
for $V(-\lambda)$.
For fixed $i \in I$ again,
can be made into a graded
2-representation of the even or reduced odd $\sl_2$ 2-category
exactly as above.
Since we are now in an Abelian setting, it makes sense
to consider $D^b(R^\lambda\smod)$,
graded super analog of the usual bounded derived category
of an Abelian category. Formally, this is defined to be the localization of $K^b(R^\lambda\smod)$ at even degree 0 quasi-isomorphisms.
The graded superequivalence of homotopy categories
obtained as in \cite[Th.~5.18]{Rou}, or our \cref{bigt} in the case that $i$ is odd, immediately implies the

\begin{theorem}\label{dereq}
In the above setup, for $\alpha \in Q^+$ such that $V(-\lambda)_{\alpha-\lambda} \neq 0$,
the even or
odd Rickard complex $\Theta_{\langle h_i, \lambda-\alpha\rangle}$
induces a graded superequivalence
\rightarrow D^b\big(R^\lambda_{\alpha-\langle h_i, \alpha-\lambda\rangle \alpha_i}\smod\big)
with quasi-inverse induced by the right adjoint
$\Theta^{\langle h_i,\lambda-\alpha\rangle}$ of this complex.
\end{theorem}

For a graded superalgebra $A$, we write $A \otimes C_1$ for the graded
superalgebra obtained by tensoring with the rank one Clifford superalgebra
generated by an odd degree 0 involution.
There is also a variation of \cref{dereq} with
$R^\lambda\smod$ replaced by
\begin{equation}
R^\lambda\otimes C_1 \smod := \bigoplus_{\alpha \in Q^+}
R^\lambda_\alpha\otimes C_1\smod.
\end{equation}
This can be made into a graded 2-representation which also categorifies
the dual Kostant $\Z[q,q^{-1}]^\pi$-form
for $V(-\lambda)$, just as $R^\lambda\smod$ did earlier.
In particular, for each $i \in I$,
we can make
$R^\lambda\otimes C_1 \smod$ into a graded
2-representation of the even or reduced odd $\sl_2$ 2-category
exactly as above. This follows by the construction explained in the next paragraph.

There is a general notion of the
{\em Clifford twist} $\cA^\CT$ of a graded
supercategory $\cA$, which goes back to \cite[Lem.~2.3]{KKT}.
By definition, this is the graded supercategory
whose objects are pairs $(X, \phi)$ for $X \in \cA$ and an
odd degree 0 involution $\phi \in \End_{\cA}(X)$.
A morphism $f:(X,\phi) \rightarrow (Y,\theta)$
is a morphism $f:X\rightarrow Y$ in $\cA$ such that
$\theta \circ f = (-1)^{\parity(f)} \phi \circ f$.
Degree and parity of morphisms in $\cA^\CT$ are induced by the ones for $\cA$.
There are obvious ways to define the
Clifford twist $F^\CT:\cA^\CT \rightarrow \cB^\CT$
of a graded superfunctor $F:\cA \rightarrow \cB$,
and also the Clifford twist $\alpha^\CT:F^\CT \Rightarrow G^\CT$
of a graded supernatural transformation $\alpha:F \Rightarrow G$
between two graded superfunctors.
This makes $\CT$ into a strict graded 2-superfunctor
$\CT:\GSCAT\rightarrow \GSCAT$.
Now if  $\cV$ is any graded 2-representation of the even or the reduced odd
2-supercategory $\UU(\sl_2)$, its Clifford twist $\cV^\CT$ can be made into a graded 2-representation
in its own right, with the required
graded superfunctors $E$ and $F$ on $\cV^\CT$ being the Clifford twists of the ones
for $\cV$, and all of the required graded supernatural transformations being the Clifford twists of the one for $\cV$ too. If $\cV$ is integrable and Karoubian then so is $\cV^\CT$.

Now the same general machinery used to prove \cref{dereq} gives the following variation.

\begin{theorem}\label{dereq2}
In the above setup, for $\alpha \in Q^+$ such that $V(-\lambda)_{\alpha-\lambda} \neq 0$,
the even or
odd Rickard complex $\Theta_{\langle h_i, \lambda-\alpha\rangle}$
induces a graded superequivalence
D^b\big(R^\lambda_{\alpha}\otimes C_1\smod\big)
\rightarrow D^b\big(R^\lambda_{\alpha-\langle h_i, \alpha-\lambda\rangle \alpha_i}\otimes C_1\smod\big)
with quasi-inverse induced by the right adjoint
$\Theta^{\langle h_i,\lambda-\alpha\rangle}$ of this complex.
\end{theorem}

\iffalse
In the sequel, we need to work in a setting where gradings are not present.
Several basic definitions need to be modified to make sense of this.
For a superalgebra $A$ (no grading!) we write
$A\realsmod$ for the supercategory of finite-dimensional left $A$-supermodules,
$\Ch^b(A\realsmod)$ for the supercategory of bounded chain complexes (even differentials but not necessarily even chain maps),
$K^b(A\realsmod)$ for the homotopy supercategory obtained from this by
identifying chain maps that are homotopic via an even homotopy,
and $D^b(A\realsmod)$ for the derived supercategory obtained
by localizing at even quasi-isomorphisms.

\begin{corollary}\label{wine}
For $\alpha \in Q^+$ such that $V(-\lambda)_{\alpha-\lambda} \neq 0$,
the even or odd Rickard complex $\Theta_{\langle h_i, \lambda-\alpha\rangle}$
induces a superequivalence
\rightarrow D^b\big(R^\lambda_{\alpha-\langle h_i, \alpha-\lambda\rangle \alpha_i}\realsmod\big)
with quasi-inverse induced by the right adjoint
$\Theta^{\langle h_i,\lambda-\alpha\rangle}$ of this complex
from the proof of \cref{bigt}.
\end{corollary}

\begin{proof}
The graded superequivalence in \cref{dereq} is obtained from
the complex of graded superfunctors
defined by the odd Rickard complex,
and its quasi-inverse is obtained from the
right adjoint of this complex. Now we are forgetting the gradings,
viewing these merely
as complexes of superfunctors. Since they are quasi-inverse with gradings present,  they are obviously quasi-inverse without these gradings, and the corollary follows.
\end{proof}
\fi

Assume henceforth that
the ground field $\k$ is algebraically closed of odd characteristic $p = 2l+1$,
and that the Cartan superdatum fixed above is of type
$A_{2l}^{(2)}$, with the shortest simple root $\alpha_0$ being odd and all other simple roots being even.
We consider the cyclotomic quiver Hecke superalgebras $R^\lambda_\alpha$
for $\alpha \in Q_+$ and $\lambda := \Lambda_0$,
taking the ring $\Bbbk$
to be the ground field $\k$, and all other choices made as
explained in \cite[Sec.~3.1]{KLi}.
$R^\lambda_\alpha \otimes C_1$
be the superalgebra tensor product of $R^\lambda_\alpha$ with the
rank one Clifford superalgebra generated by an odd involution.
Now we forget both the $\Z$- and $\Z/2$-gradings
on $R^\lambda_\alpha$ and $R^\lambda_\alpha \otimes C_1$
to view them as ordinary finite-dimensional algebras.
For such an algebra $A$, we write $A\fdmod$ for the Abelian category
of finite-dimensional left $A$-modules and $D^b(A\fdmod)$ for its ordinary
bounded derived category.

In view of \cite[Lem.~3.1.39]{KLi},
the following proves \cite[Conj.~2]{KLi}.

\begin{theorem}
Suppose that $\alpha, \beta \in Q^+$ are
such that $\alpha-\lambda$ and $\beta - \lambda$ are weights of
$V(-\lambda)$ in the same $W$-orbit.
The categories
$D^b\big(R^\lambda_\beta\fdmod\big)$ are equivalent
as are $D^b\big(R^\lambda_\alpha\otimes C_1\fdmod\big)$
$D^b\big(R^\lambda_\beta\otimes C_1 \fdmod\big)$.
\end{theorem}

\begin{proof}
Since the simple reflections generate $W$,
it suffices to prove the theorem in the special case that
$\alpha-\lambda$ is a weight of $V(-\lambda)$
and $\beta = \alpha-\langle h_i, \alpha-\lambda\rangle \alpha_i$ for some $i \in I$.
The graded superequivalences in \cref{dereq,dereq2} are obtained by taking the derived tensor product
with the complex of graded superbimodules
arising from the appropriate Rickard complex.
Similarly, the quasi-inverse graded superequivalences are obtained from the
right adjoint of this complex. Now we are forgetting both the $\Z$- and $\Z/2$-gradings,
viewing these complexes of graded superbimodules as complexes of ordinary bimodules.
The resulting complexes define functors
between the ordinary derived categories.
Since they are quasi-inverse with all gradings present, they are obviously quasi-inverse without these gradings.
\end{proof}

\begin{corollary}\label{CBroue}
Brou\'e's Abelian Defect Group Conjecture holds for double covers of symmetric and alternating groups over any algebraically closed field of positive characteristic.
\end{corollary}

\begin{proof}
See \cite[Th.~5.4.12]{KLi} where this is deduced from \cite[Conj.~2]{KLi}.
\end{proof}

\iffalse
Let $\hat S_n$ be a double cover of the symmetric group with canonical central
element $z$.
Setting $e_{\pm} := \frac{1}{2}(1\pm z) \in \k \hat S_n$, we have that
\k \hat S_n = e_+ \k \hat S_n \oplus e_- \k \hat S_n
with $e_+ \k \hat S_n \equiv \k S_n$.
We are interested in {\em spin representations} of $\k \hat S_n$, that is,
representations of the algebra $S(n) := e_- \k \hat S_n$.
We view $S(n)$ as a superalgebra so that the image in $S(n)$
of a group element $g \in S_n$ is even or odd according to the parity of the permutation $g$.
Apart from the case of trivial defect, for which Brou\'e's conjecture is trivial,
the blocks (indecomposable summands) of the algebra $S(n)$
are the same as its indecomposable summands viewed as a superalgebra,
i.e., they are {\em superblocks}.
As explained in [4], superblocks of $S(n)$ are parametrized by the
weight spaces of the basic representation $V(-\lambda)$
of the Kac-Moody algebra of type $A_{2l}^{(2)}$.
We denote the superblock correspond to the weight space
$V(\lambda)_{\alpha-\lambda}$ by $B_\alpha$.
Moreover, it is known that any superblock of $S(n)$
is either of type $\mathtt{M}$ or $\mathtt{Q}$ according to the common
type (in the sense of the superalgebra analog of Schur's Lemma) of
irreducible supermodules for the superblock.
In fact, assuming $\alpha = \sum_{i \in I} c_i \alpha_i$,
$B_\alpha$ is of type $\mathtt{M}$
if $c_1+\cdots+c_\ell$ is even
and of type $\mathtt{Q}$ otherwise;
see \cite[Th.~9.10]{BK}.

The results of [26] show that
$B_\alpha$ is Morita superequivalent to
$R^\lambda_\alpha$ if $B_\alpha$ is of type $\mathtt{M}$
or the superalgebra tensor product
$R^\lambda_\alpha \otimes C_1$ if $B_\alpha$ is of type $\mathtt{Q}$,
where $C_1$ here is the rank one Clifford superalgebra
generated by an odd involution $c$; cf. \cite[Th.~5.3.31]{KLi}.

Letting $W$ be the Weyl group of this type,
$W$-orbits of weight spaces of $V(-\Lambda_0)$ are naturally labelled by natural numbers,
also known the {\em $p$-bar weight} of the corresponding superblock.
Also, the defect group associated
to the superblock is Abelian if and only its $p$-bar weight $d$
satisfies $d < p$; we also assume $d > 0$ to avoid making exceptions in the case of
trivial defect.

In [29], the notion of a {\em RoCK superblock} of $S(n)$ is introduced as
an analog of the notion of RoCK block for ordinary represtations of symmetric groups.
It is shown in \cite[Theorem B]{KLi}
that any RoCK superblock with $p$-bar weight $0 < d < p$
is derived equivalent to its Brauer
correspondent. Moreover all superblocks of fixed $p$-bar weight $0 < d < p$
and fixed type ($\mathtt{M}$
or $\mathtt{Q}$) have the same Brauer correspondent up to Morita
Bearing in mind that there exist RoCK superblocks of every $p$-bar weight and type,
this reduces the proof of Brou\'e's conjecture for spin representations of $S_n$ is reduced
to showing that any superblock of $p$-bar weight $0 < d < p$ is derived superequivalent
to a RoCK superblock of the same $p$-bar weight and type.

Finally, it is shown in [26] that the superblock $B(\alpha)$
of $S(n)$
associated to the weight space $V(-\Lambda_0)_{\alpha-\Lambda_0}$
is Morita superequivalent to the (undeformed) cyclotomic quiver Hecke
superalgebra $\overline{R}^{\Lambda_0}_\alpha$ if $B(\alpha)$ is of
type $\mathtt{M}$, or to its ``Clifford twist"
$\overline{R}^{\Lambda_0}_\alpha \otimes C_1$
where $C_1$ is a rank 1 Clifford superalgebra
if $B(\alpha)$ is of type $\mathtt{Q}$.
\cref{dereq} implies that
and $\overline{R}^{\Lambda_0}_\beta$
are derived superequivalent for $\alpha,\beta \in Q^+$
with $\alpha-\Lambda_0$ and $\beta - \Lambda_0$ in the same $W$-orbit.
The following now follows.

\begin{theorem}
Any two superblocks of $S(n)$ of the same $p$-bar weight and type are
derived superequivalent.
\end{theorem}

\begin{corollary}\label{CBroue}\
Brou\'e's Abelian Defect Group Conjecture holds for double covers of symmetric and alternating groups in odd characteristic $p > 2$.
\end{corollary}

\begin{proof}
The discussion above plus the known result for ordinary blocks of symmetric groups
established in [7]
completes the proof for the double covers of symmetric groups.
The result for ordinary blocks of alternating groups
is established by Marcus [33]
in all odd characteristics, by reducing to symmetric groups.
Finally, the result for spin blocks of alternating groups
follows from the result for spin blocks of symmetric groups
together with [29], which shows that any spin block of $\widehat{A}_n$
is Morita equivalent to the Clifford twist of a corresponding block of
$\widehat{S}_n$, and [29] which shows that the same is true for the Brauer correspondent.
\end{proof}
\fi

\iffalse
We now adopt the notation of [29].

In particular, ${\mathfrak g}$ is the Kac-Moody Lie algebra of type $A_{2\ell}^{(2)}$ with the simple roots $\{\alpha_i\mid i\in I\}$ labeled by $I:=\{0,1,\dots,\ell\}$, the corresponding non-negative part of the root lattice $Q^+=\sum_{i\in I}\Z_{\geq 0}\cdot \alpha_i$, the fundamental dominant weights $\{\Lambda_i\mid i\in I\}$, and the Weyl group $W$ generated by the fundamental reflections $\{r_i\mid i\in I\}$. The Weyl group $W$ naturally acts on the weight lattice $P=\bigoplus_{i\in I}\Z\cdot\Lambda_i$.

We denote by $R_\theta^{\Lambda_0}$ the cyclotomic quiver Hecke superalgebra corresponding to $\theta\in Q^+$ and the dominant weight $\Lambda_0$.

\begin{theorem}\label{TC2}\
Let $\theta,\eta\in Q^+$ be such that $\Lambda_0-\theta$ is in the same $W$-orbit as $\Lambda_0-\eta$. Then the superalgebras $R_\theta^{\Lambda_0}$ are derived equivalent. In particular, \cite[Conjecture 2]{KLi} holds.
\end{theorem}

\begin{proof}
Let $p=2\ell+1$. By \cite[Lemma 3.1.39]{KLi}, the condition that $\Lambda_0-\theta$ is in the same $W$-orbit as $\Lambda_0-\eta$ is equivalent to the condition that $\theta$ and $\eta$ are of the forms  $\theta=\operatorname{cont}(\rho)+d\delta$ and $\eta=\operatorname{cont}(\rho')+d\delta$ for some $\bar p$-cores $\rho,\rho'$ and $d\in\Z_{\geq 0}$. So \cite[Conjecture 2]{KLi} does follow from the main claim of the theorem.

To prove the main claim, since $W$ is generated by the fundamental reflections $r_i$, we may assume that $\Lambda_0-\eta=r_i(\Lambda_0-\theta)$ for some $i\in I$.
\end{proof}
\fi

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