We first show the case when $Y=\Gamma_{rel}$ and $i\leq 1$.

Recall the relative t-structure $(\D(\Gamma_{rel})^{rel}_{\leq 0}, \D(\Gamma_{rel})^{rel}_{\geq 0})$ on $\D(\Gamma_{rel})$ which restricts to t-structures on $\per \Gamma_{rel}$ and on $\pvd_{e}(\Gamma_{rel})$, cf. <cit.>.

We show the injectivity of the map.
Let $f:X\rightarrow \Sigma^{i}\Gamma_{rel}$ be a map in $\per(\Gamma_{rel})$ which factors through an object $M$ in $\pvd_{e}(\Gamma_{rel})$.

We want to show that $f$ is indeed a zero map.
By definition $\Gamma_{rel}\in \per(\Gamma_{rel})^{rel}_{\leq 0}$ and hence $X\in \add \Gamma_{rel}*\add \Sigma\Gamma_{rel}\subset \per(\Gamma_{rel})^{rel}_{\leq 0}$.
So we may suppose $M\in \pvd_{e}(\Gamma_{rel})^{rel}_{\leq 0}$.

By <cit.>, we have

\[
\Hom_{\Gamma_{rel}}(M,\Sigma^i\Gamma_{rel})\xrightarrow{\sim}D\Hom_{\Gamma_{rel}}(\Gamma_{rel}, \Sigma^{3-i}M)=0
\]

where $D=\Hom(-,k)$.

This shows the injectivity of the morphism.
We now show the surjectivity of the map. Consider a roof $b/s$

\[
\begin{tikzcd}
\end{tikzcd}.
\]

Since $X\in\per(\Gamma_{rel})^{rel}_{\leq 0}$, we may assume $\Cone(s)\in \pvd(\Gamma_{rel})^{rel}_{\leq 0}$.
Then since we have

\[
\Hom_{\Gamma_{rel}}(\Sigma^{-1}\Cone(s),\Sigma^{i} \Gamma_{rel})\iso D\Hom_{\Gamma_{rel}}(\Gamma,\Sigma^{2-i} \Cone(s))=0,
\]

the morphism $b$ factors through $s$ and this shows the surjectivity of the map.

For general $Y\in \mathcal F^{rel}$ and $i\leq 0$, it follows from the above case by applying the homological functor $\Hom(X,-)$ to the defining triangle for $Y$.

Let $\per_{dg}(\Gamma_{rel})$ (resp. $\pvd_{e}(\Gamma_{rel})_{dg}$) be the canonical dg enhancement of $\per(\Gamma_{rel})$ (resp. $\pvd_{e}(\Gamma_{rel})$).
Let $\C_{dg}$ be the canonical dg enhancement of $\C=\C(Q,F,W)$ given by the dg quotient $\per_{dg}(\Gamma_{rel})/\pvd_{e}(\Gamma_{rel})_{dg}$.
Let $\mathcal H_{dg}$ be the full dg subcategory of $\C_{dg}$ consisting of objects in $\mathcal H$.
Let $\F^{rel}_{dg}$ be the dg full subcategory of $\per_{dg}(\Gamma_{rel})$ consisting of objects in $\F^{rel}$.

By <cit.>, the dg category $\F^{rel}_{dg}$ (resp. $\mathcal H_{dg}$) is extension-closed in $\per_{dg}(\Gamma_{rel})$ (resp. $\C_{dg}$) and thus inherits a canonical exact dg structure.
By Lemma <ref>, the canonical morphism from $\per_{dg}(\Gamma_{rel})$ to $\C_{dg}$ induces a quasi-equivalence

\[
\tau_{\leq 0}\F_{dg}^{rel}\rightarrow \tau_{\leq 0} \mathcal H_{dg}.
\]

By <cit.>, Theorem <ref> and Proposition <ref>, the canonical morphism

\[
\tau_{\leq 0}\mathcal H_{dg}\rightarrow \C_{dg}
\]

characterises $\C_{dg}$ as the bounded dg derived category of $\tau_{\leq 0}\mathcal H_{dg}$.

Since $\Gamma_{rel}$ is projective in the extriangulated category $\F^{rel}$, it follows that higher extensions $\mathbb E^i$ of $\F^{rel}$ vanish for $i\geq 2$.
Also we have $\Ext^{i}_{\per(\Gamma_{rel})}(X,Y)=0$ for $X,Y\in\F^{rel}$ and $i\geq 2$.
Thus by  Theorem <ref> and Proposition <ref>, the canonical morphism

\[
\tau_{\leq 0}\mathcal F^{rel}_{dg}\rightarrow \per_{dg}(\Gamma_{rel})
\]

characterises $\per_{dg}(\Gamma_{rel})$ as the bounded dg derived category of $\tau_{\leq 0}\mathcal F^{rel}_{dg}$.

Claim: Via the canonical quasi-equivalence $\tau_{\leq 0}\F^{rel}_{dg}\rightarrow \tau_{\leq 0} \mathcal H_{dg}$, the exact dg structure on $\tau_{\leq 0}\F^{rel}_{dg}$ is identified with the exact substructure on $\tau_{\leq 0}\mathcal H_{dg}$ left generated by $\add \Gamma_{rel}$.

Indeed, let $g:\pi^{rel}X\rightarrow \Sigma \pi^{rel}Y$ be a morphism in $\mathcal H$.
Then by Lemma <ref>, $g$ is of the form $\pi^{rel}(f)$ for some $f:X\rightarrow \Sigma Y$ if and only if $g$ factors though an object in $\add \Gamma_{rel}$.

Note that $\Gamma_{rel}$ is cluster-tilting in $\mathcal H$, cf. <cit.>. In general we have

Let $(\A,\mathcal S)$ be a connective exact dg category such that $(H^0(\A),\mathbb E)$ is a Hom-finite, Krull–Schmidt extriangulated category with a basic cluster-tilting object $T$.
Let $\Gamma$ be the dg endomorphism algebra of $T$ in $\A$.
Let $(\A,\mathcal S')$ be the connective exact dg category with exact substructure left generated by $\add T$.
Let $\D^b_{\mathcal S'}(\A){\coloneqq}H^0(\D^b_{dg}(\A))$ be the bounded derived category of $\A$ with the exact structure $\mathcal S'$.
Then we have an equivalence of triangulated categories up to direct summand

\[
\D^b_{\mathcal S'}(\A)\rightarrow \per\Gamma.
\]

The dg functor

\[
\Hom(T,-):\D^b_{dg}(\A)\rightarrow \C_{dg}(\Gamma)
\]

induces a functor $D^b_{\mathcal S'}(\A)\rightarrow \per\Gamma$.
Notice that $T$ is projective in $(H^0(\A),\mathbb E')$ with $\mathbb E'$ given by the exact structure $\mathcal S'$.

Since $T$ is cluster-tilting, the extriangulated category $(H^0(\A),\mathbb E')$ has enough projective objects $\add T$ and is hereditary in the sense of [53].
Also, $T$ generates $\D^b_{\mathcal S'}(\A)$ as a triangulated category.

Now the proposition follows from Theorem <ref> and Proposition <ref>, by a dévissage argument, cf. <cit.><cit.>.

In Proposition <ref>, if we take different basic cluster-tilting objects $T$ in $(H^0(\A),\mathbb E)$, then the exact substructures thus constructed are different because they have different projective objects.

We end this section by constructing a class of homotopy bicartesian squares in the dg category $\mathcal F^{rel}_{dg}$ (or, by Remark <ref>, equivalently in the dg category $\tau_{\leq 0}\F^{rel}_{dg}$) which are not necessarily homotopy bicartesian in the ambient pretriangulated dg category $\per_{dg}(\Gamma_{rel})$.
The idea is that we have a quasi-equivalence of dg categories

\[
\tau_{\leq 0}\F^{rel}_{dg}\simeq\tau_{\leq 0}\mathcal H_{dg}
\]

which identifies the exact structure on the left hand side with a substructure of that on the right hand side.
So we only need to find conflations in $\tau_{\leq 0}\mathcal H_{dg}$ which are not in $\tau_{\leq 0}\F^{rel}_{dg}$.

Let $M$ be an object in $\mathcal F^{rel}$ of the form $\Cone(P\xrightarrow{f}Q)$.
Denote by $M^{*}$ the dual $\RHom_{\Gamma_{rel}}(M,\Gamma_{rel})\in \per(\Gamma^{op}_{rel})$ of $M$.
Then we have $H^iM^{*}=0$ for $i\geq 2$ and that $H^1M^{*}$ is finite-dimensional.
Thus $N=\tau_{\leq 0}M^{*}$ lies in $\per(\Gamma^{op}_{rel})$.

Claim 3: We have $H^1M^{*}\in \pvd_{e}(\Gamma^{op}_{rel})$ and $M\xrightarrow{\sim} \tau_{\leq 0}N^{*}$.

Indeed, we apply the cohomological functor $\Hom(-,\Gamma_{rel})$ to the triangle

\[
P\xrightarrow{f}Q\rightarrow M\rightarrow \Sigma P
\]

and we get the following exact sequence

\[
\begin{tikzcd}
\Hom(\Sigma Q, \Sigma e\Gamma_{rel})\ar[r, two heads]&\Hom(\Sigma P,\Sigma e\Gamma_{rel})\ar[r]&\Hom(M,\Sigma e\Gamma_{rel})\ar[r]&\Hom(Q,\Sigma e\Gamma_{rel})=0.
\end{tikzcd}
\]

Thus we have $eH^1M^*=\Hom(M,\Sigma e\Gamma_{rel})=0$ and $H^1M^*\in \pvd_{e}(\Gamma^{op}_{rel})$.
We have a canonical triangle

\[
\begin{tikzcd}
(\Sigma^{-1}H^1M^{*})^{*}\ar[r]&M^{**}\ar[r]& N^{*}\ar[r]&\Sigma(\Sigma^{-1}H^1M^*)^*.
\end{tikzcd}
\]

Then the claim follows by observing $(H^1M^*)^*\in \D(\Gamma^{op}_{rel})_{\geq 3}$.
Indeed, we have

\[
(H^1M^*)^*\in \D(\Gamma^{op}_{rel})_{\geq 3}\cap\D(\Gamma^{op}_{rel})_{\leq 3}
\]

which implies that $N^{*}\in \D(\Gamma_{rel}^{op})_{\leq 1}^{rel}$.

Claim 4: We have $\Hom(N,\Sigma^{i} \Gamma_{rel}e)\xrightarrow{\sim}H^i(M)e=0$ for $i\geq 1$ and $\Hom(N,T)\xrightarrow{\sim} \Hom(H^1M^{*}, \Sigma^2 T)$ for $T\in \D(\Gamma^{op}_{rel})_{\leq -2}$.

Indeed, the claim follows by applying the cohomological functors $\Hom(-,\Sigma \Gamma_{rel}e)$ and $\Hom(-,T)$ to the triangle in $\per(\Gamma_{rel}^{op})$

\[
N=\tau_{\leq 0}M^{*}\rightarrow M^{*}\rightarrow \Sigma^{-1}H^1M^{*}\rightarrow \Sigma N.
\]

We also have $\Hom(\Gamma_{rel}e,\Sigma^{\geq 1}N)=0$.

Summarising the above discussion, we have, by <cit.>, that $N\in \F^{rel}_{\Gamma^{op}_{rel}}$ if and only if $\Hom(H^1M^{*},T)=0$ for any $T\in \D(\Gamma^{op}_{rel})_{\leq -4}$.

Keep the notations as above.
Suppose that $N$ lies in $\mathcal F^{rel}_{\Gamma^{op}_{rel}}$ and has a defining triangle

\[
\begin{tikzcd}
R\ar[r,"g"] &S\ar[r]&N\ar[r]&\Sigma R
\end{tikzcd}
\]

where $R$ and $S$ belongs to $\add\Gamma_{rel}$ and the morphism $g$ induces a surjection

\[
\Hom_{\Gamma^{op}_{rel}}(S,e\Gamma^{op}_{rel})\rightarrow \Hom_{\Gamma^{op}_{rel}}(R,e\Gamma^{op}_{rel}).
\]

Then the following diagram in $\F^{rel}\subset \D(\F^{rel}_{dg})$

\begin{equation}\label{diagram}
\begin{tikzcd}
0\ar[r]& R^*
\end{tikzcd},
\end{equation}

where the map $u$ is given by

\[
M\xrightarrow{\sim}\tau_{\leq 0}N^*\rightarrow N^*\rightarrow S^{*},
\]

lifts to an object in $\rep(\mathrm{Sq},\F^{rel}_{dg})$, which is homotopy bicartesian, under the canonical functor $\Dia:\D((k\mathrm{Sq})^{op}\otimes \F^{rel}_{dg})\rightarrow \Fun(\mathrm{Sq}, \D(\F^{rel}_{dg}))$.
Moreover, it is homotopy bicartesian with respect to $\per_{dg}(\Gamma_{rel})$ if and only if $\Hom(M,\Sigma\Gamma_{rel})=0$.

We may assume $N^*$ is strictly concentrated in degrees $\leq 1$ and $M=\tau_{\leq 0} N^{*}$.
We also assume the triangle

\[
N^{*}\rightarrow S^*\rightarrow R^*\rightarrow \Sigma N^*
\]

is induced by a short exact sequence of dg $\Gamma^{op}_{rel}$-modules

\[
0\rightarrow N^*\rightarrow S^*\rightarrow R^*\rightarrow 0.
\]

Then we have the following commutative diagram in $\C(\Gamma_{rel}^{op})$

\[
\begin{tikzcd}
M\ar[r,tail]\ar[d,equal] &N^*\ar[r,two heads]\ar[d,tail]& \Sigma^{-1} H^1N^*\ar[d,tail]\\
M\ar[r,tail]&S^*\ar[r, two heads]\ar[d,two heads]&U\ar[d,two heads]\\
&R^*\ar[r,equal]& R^*
\end{tikzcd}.
\]

Since $\Hom(H^1N^*,\Sigma^{\leq 1}X)=0$ and $\Hom(X,\Sigma^{\leq -1}H^1N^*)=0$ for $X\in \F^{rel}_{\Gamma_{rel}^{op}}$, the following object in $\rep(\mathrm{Sq},\F^{rel}_{\Gamma^{op}_{rel}})$

\[
\begin{tikzcd}
\end{tikzcd}
\]

is a lift of the diagram <ref> which is homotopy bicartesian.

Let $(Q,F,W)$ be the following ice quiver with potential

\[
\begin{tikzcd}
\textcolor{blue}3\ar[ru,"\alpha"]&&\textcolor{blue}1\ar[ld,"\gamma"blue, blue]\\
&\textcolor{blue}2\ar[lu,"\delta"blue, blue]&
\end{tikzcd}
\]

where the ice part $F$ is given by the blue vertices and blue arrows and the potential $W=\delta\gamma\beta\alpha$.
Then the underlying graded quiver of the corresponding relative Ginzburg algebra $\Gamma_{rel}(Q,F,W)$ is given as follows

\[
\begin{tikzcd}
&4\ar[rd,"\beta", shift left=1ex]\ar[loop, looseness=4, "t_4"swap, green]\ar[ld,"\alpha^{*}",shift left=1ex,red]&\\
\textcolor{blue}3\ar[ru,"\alpha", shift left =1ex]&&\textcolor{blue}1\ar[ld,"\gamma"blue, blue]\ar[lu,"\beta^{*}", shift left=1ex,red]\\
&\textcolor{blue}2\ar[lu,"\delta"blue, blue]&
\end{tikzcd}
\]

where $|\alpha^*|=-1$, $|\beta^{*}|=-1$ and $|t_4|=-2$. The differential $d$ is determined by the values on generators, which are given as follows

\begin{gather*}
d(\alpha^*)=\delta\gamma\beta, d(\beta^*)=\alpha\beta\gamma, d(t_4)=\alpha\alpha^*-\beta^*\beta.
\end{gather*}

We have $e=e_1+e_2+e_3$. Denote by $P_i= e_i\Gamma_{rel}$.

Let $M=S_1$. Then we have the following triangle

\[
P_4\rightarrow P_1\rightarrow S_1\rightarrow \Sigma P_4
\]

such that

\[
\Hom(P_1, e\Gamma_{rel})\twoheadrightarrow \Hom(P_4,e\Gamma_{rel}).
\]

So $M\in \F^{rel}$.

Let $L$ be the homotopy fiber of the map $P_3\rightarrow P_4$ induced by $\alpha$.
Then we have a triangle

\[
D\rightarrow L\rightarrow \Sigma^{-1} S_4\rightarrow \Sigma D
\]

where $D$ is the mapping cone of the map $P_4\rightarrow P_1$ induced by $\beta$, which is isomorphic to $S_1$.

So $N=\tau_{\leq 0} M^*$ is isomorphic to $S_3'$.
Similarly we have the following triangle

\[
P_4^*\rightarrow P_3^*\rightarrow S_3'\rightarrow \Sigma P_4^*
\]

which shows that $N\in \F^{rel}$.

We thus have the following diagram in $\F^{rel}$

\[
\begin{tikzcd}
S_1\ar[r]\ar[d]& P_3\ar[d]\\
0\ar[r] &P_4
\end{tikzcd}
\]

which can be lifted to a homotopy bicartesian square in $\F^{rel}_{dg}$.

§ THE GREATEST EXACT STRUCTURE ON AN ADDITIVE DG CATEGORY

We follow Rump's idea [125, 126] to show the existence of the greatest exact structure.

Let $\A$ be an additive connective dg category. For simplicity, we may assume that $Z^0(\A)$ is an additive category, cf. Remark <ref>.

A left exact structure on $\A$ is a class $\mathcal{S}\subseteq \mathcal H_{3t}(\A)$ stable under isomorphisms, consisting of homotopy short exact sequences (called conflations)

\[
\begin{tikzcd}
A\ar[r, tail,"i"]\ar[rr,bend right=8ex,"h"swap]&B\ar[r,two heads, "p"]&C\\
\end{tikzcd}
\]

where $i$ is called an inflation and $p$ is called a deflation, such that the following axioms are satisfied

Ex0$\Id_{0}$ is a deflation.

Ex1Compositions of deflations are deflations.

Ex2Given a deflation $p:B\rightarrow C$ and any map $c: C'\rightarrow C$ in $Z^0(\A)$, the object

\[
\begin{tikzcd}
\end{tikzcd}
\]

admits a homotopy pullback

\[
\begin{tikzcd}
\end{tikzcd}
\]

and ${p'}$ is also a deflation.

Ex3If a morphism $g:B\rightarrow C$ admits a homotopy kernel and for some $h:B'\rightarrow B$ the composition $gh:B'\rightarrow C$ is a deflation, then $g$ is a deflation.

We call $(\A,\mathcal {S})$ or simply $\A$ a left exact dg category.

Dually, a right exact structure on $\A$ is given by a class of homotopy short exact sequences which satisfies the axioms ${\Ex0}$, ${\Ex1}^{op}$, ${\Ex2}^{op}$ and ${\Ex3}^{op}$.
By the axioms $\Ex0$ and $\Ex2$, the morphism $\Id_A$ is a deflation for each $A\in\A$.
By the axioms $\Ex0$ and $\Ex3$, the morphism $A\rightarrow 0$ is a deflation for each $A\in\A$.
By Axiom $\Ex3$, in the context of Axiom $\Ex2$, the 3-term h-complex

\[
\begin{tikzcd}
B'\ar[r,"\begin{bmatrix}b\\p'\end{bmatrix}"]\ar[rr,bend right=8ex,"s"swap]&B\oplus C'\ar[r,"{[}-p{,}\;c{]}"]&C
\end{tikzcd}
\]

is a conflation.

Let $\A$ be an additive connective dg category equipped with a left exact structure $\mathcal{S}_1$ and a right exact structure $\mathcal{S}_2$.
Then the class of homotopy short exact sequences

\[
\begin{tikzcd}
A\ar[r, tail,"f"]\ar[rr,"h"swap,bend right=8ex]&B\ar[r,two heads, "j"]&C
\end{tikzcd}
\]

where $f$ is an inflation in $\mathcal S_2$ and $j$ is a deflation in $\mathcal S_1$, makes $\A$ into an exact dg category.

Consider a homotopy short exact sequence $X$

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,"h"swap,bend right=8ex]&B\ar[r,"j"]&C
\end{tikzcd}
\]

where $j$ is a deflation in $\mathcal S_1$ and $f$ is an inflation in $\mathcal S_2$.
We call such homotopy short exact sequences conflations and denote by $\mathcal S$ the class of conflations.
We will show that the class $\mathcal S$ defines an exact dg structure on $\A$.
It is clear that $\mathcal S$ is closed under isomorphisms in $\mathcal H_{3t}(\A)$.
Let us point out that the axioms $\Ex2$ and $\Ex2^{op}$ can be verified directly, using Axiom $\Ex3$ for $\mathcal S_1$ and Axiom $\Ex3^{op}$ for $\mathcal S_2$.
So it remains to show Axiom $\Ex1$ for $\mathcal S$.

Let $X'$ be another conflation

\[
\begin{tikzcd}
E\ar[r,"f'"]\ar[rr,"h'"swap,bend right=8ex]&C\ar[r,"j'"]&F\mathrlap{.}
\end{tikzcd}
\]

We want to show that the composition $j'j$ remains a deflation in $\mathcal S$.
Note that Lemma <ref> remains true for the dg category $\A$ with the class $\mathcal S$ of homotopy short exact sequences.
The composition $j'j$ is a deflation in $\mathcal S_1$, so it admits a homotopy kernel $\tilde{X'}$

\[
\begin{tikzcd}
{D}\ar[r,"{f''}"]\ar[rr,"h''"swap, bend right=8ex]&B\ar[r,"j'j"]&F
\end{tikzcd}
\]

which is homotopy short exact.
By Lemma <ref>, the obvious morphism from $B\xrightarrow{j'j} F$ to $C\xrightarrow{j'} F$ in $H^0(\Mor(\A))$ induces a morphism $\mu:\tilde{X'}\rightarrow X'$ in $\mathcal H_{3t}(\A)$

\[
\begin{tikzcd}
D\ar[rr,bend left=8ex,"h''"]\ar[rrd,"t"{blue},blue]\ar[d,"l"swap]\ar[rd,"s_2"{red,swap},red]\ar[r,"f''"]&B\ar[rd,"0"red,red]\ar[d,"{j}"swap]\ar[r,"j'j"]&F\ar[d,equal]\\
E\ar[rr,bend right=8ex,"h'"swap]\ar[r,"f'"swap]&C\ar[r,"j'"swap]&F
\end{tikzcd}
\]

which restricts to a homotopy pullback $Y$ of the cospan $L$

\[
\begin{tikzcd}
\end{tikzcd}
\]

Apply Axiom ${\Ex2}$ to the cospan $L$ and we get a morphism $\theta:\tilde{X}\rightarrow X$ of conflations in $\mathcal S_1$ (cf. Proposition <ref>)

\[
\begin{tikzcd}
A\ar[rd,"s_1"{red,swap},red]\ar[rrd,"t"{blue},bend left=2ex,blue]\ar[rr,bend left=8ex,"\tilde{h}"]\ar[r,"k"]\ar[d,equal]&D\ar[r,"l"]\ar[d,"f''"swap]\ar[rd,"s_2"red,red]&E\ar[d,"f'"]\\
A\ar[r, "f"swap]\ar[rr,"h"swap,bend right=8ex]&B\ar[r,"j"swap]&C
\end{tikzcd}
\]

Note that the object $Y$ is a homotopy bicartesian square, cf. the proof of Proposition <ref>.

We have the following object in $\rep(\overline{\Sq},\A)$

\[
\begin{tikzcd}
B\ar[r,"\begin{bmatrix}1\\0\end{bmatrix}"swap]&B\oplus E
\end{tikzcd}
\]

where $u=[s_1,\tilde{h}]$.
It is a graded-split extension of

\[
\begin{tikzcd}
\end{tikzcd}
\]

\[
\begin{tikzcd}
\end{tikzcd}
\]

and is thus homotopy bicartesian by Lemma <ref>.
So $\begin{bmatrix}f''\\l\end{bmatrix}:D\rightarrow B\oplus E$ is an inflation in $\mathcal S_2$.

Similarly, the following object

\[
\begin{tikzcd}
D\ar[rd,"v"]\ar[d,"\begin{bmatrix}f''\\l\end{bmatrix}"swap]\ar[r,"f''"]     &B\ar[d,"\begin{bmatrix}1\\j\end{bmatrix}"]      \\
B\oplus E\ar[r,swap,"\begin{bmatrix}1\ 0\\0\ f'\end{bmatrix}"]           &                            B\oplus C
\end{tikzcd}
\]

where $v=[0,-s_2]^{\intercal}$, is a homotopy bicartesian square.
Therefore $\begin{bmatrix}1\\j\end{bmatrix}:B\rightarrow B\oplus C$ is an inflation in $\mathcal S_2$.
Notice that $\begin{bmatrix}1&0\\0&f'\end{bmatrix}:B\oplus E\rightarrow B\oplus C$ is an inflation in $\mathcal S_2$ as a homotopy pushout of the inflation $f':E\rightarrow C$.
So $f''$ is also an inflation in $\mathcal S_2$ by ${\Ex3}^{op}$.
Therefore $j'j$ is a deflation in $\mathcal S$ and this finishes the proof.

Let $\R'$ be a class of objects $p:Y\rightarrow Z$ in $H^0(\Mor(\A))$ which is closed under isomorphisms.
The class $\R'$ is stable under pullbacks, provided that for any object in $\R'$, the cospan along any morphism in $Z^0(\A)$ admits a homotopy pullback and that $\R'$ is stable under forming homotopy pullbacks.
We define $P\R'\subset \R'$ to be the class of objects $f:C\rightarrow F$ such that for any $h:E\rightarrow F$ in $H^0(\Mor(\A))$, the cospan $L$

\begin{equation}\label{cospan:max}
\begin{tikzcd}
\end{tikzcd}
\end{equation}

admits a homotopy pullback $X$

\begin{equation}\label{pullback:max}
\begin{tikzcd}
{B}\ar[rd,"s"]\ar[d,"e"swap]\ar[r," {b}"]& {C}\ar[d,"  f"]\\
{E}\ar[r,"h"swap]& {F}
\end{tikzcd}
\end{equation}

such that ${e}:B\rightarrow E$ belongs to $\R'$.
In particular, objects in $P\R'$ admits homotopy kernels.
Note that $P\R'=\R'$ if and only if $\R'$ is stable under pullbacks.

We write $Q\R'\subset \R'$ for the class of objects $f:C\rightarrow F$ such that if a homotopy pullback $X$ as above exists, then $ {b}\in \R'$ implies $ {h}\in \R'$.
The classes $P\R'$ and $Q\R'$ are again closed under isomorphisms in $H^0(\Mor(\A))$.
Note that for an object $A\in \A$, if the object $\Id_A\in H^0(\Mor(\A))$ belongs to $\R'$, then it also belongs to the classes $P\R'$ and $Q\R'$.

Let $\R$ be the class of objects $p:Y\rightarrow Z$ in $H^0(\Mor(\A))$ which admits a homotopy kernel which is homotopy bicartesian. Clearly, the class $\R$ is closed under isomorphisms in $H^0(\Mor(\A))$ and contains $\Id_{A}:A\rightarrow A$ for $A\in\A$.
It is clear that to define a left exact structure, it is enough to specify the class $\R'$ of deflations, which is contained in $\R$, satisfying the axioms $\Ex0$–$\Ex3$.
Note that whenever $\R'$ defines a left exact structure, we have $Q\R'=\R'$.

Keep the assumptions as above. The following statements hold.

1)We have $PP(\R')=P(\R')$, i.e. $P\R'$ is stable under pullbacks;

2)If the class $\R'$ is stable under forming compositions, the same holds for $P\R'$.

1). Let $f:C\rightarrow F$ be an object in $P\R'$ and $h:E\rightarrow F$ any object in $H^0(\Mor(\A))$.
Then the cospan $L$ (<ref>) admits a homotopy pullback $X$ of the form (<ref>) such that $ {e}\in \mathcal R'$.

Claim: The object ${e}: B\rightarrow E$ belongs to $P\R'$.
Indeed, let $ {g}: {D}\rightarrow  {E}$ be any object in $H^0(\Mor(\A))$, then the cospan $L'$

\[
\begin{tikzcd}
& {C}\ar[d,"  f"]\\
{D}\ar[r,"  h  g"swap]& {F}
\end{tikzcd}
\]

admits a homotopy pullback $X'$

\[
\begin{tikzcd}
\end{tikzcd}
\]

The obvious morphism $L'\rightarrow L$ between cospans induces a morphism $X'\rightarrow X$ (cf. Lemma <ref>). Thus, by restriction we obtain a homotopy pullback square (cf. Proposition <ref>)

\[
\begin{tikzcd}
\end{tikzcd}
\]

Now apply Corollary <ref> and we see that $b':A'\rightarrow B$ belongs to $\R'$.
Thus we have ${e}\in P\mathcal R'$ and this proves 1).

2). Let $g:D\rightarrow E$ and $h:E\rightarrow F$ be two objects in $P\mathcal R'$ and $f:C\rightarrow F$ any object in $H^0(\Mor(\A))$.
Then the cospan (<ref>) admits a homotopy pullback (<ref>) where $b$ belongs to $\R'$.
The cospan

\[
\begin{tikzcd}
\end{tikzcd}
\]

also admits a homotopy pullback square

\[
\begin{tikzcd}
\end{tikzcd}
\]

where $b':A'\rightarrow B$ belongs to $\R'$ and hence the composition $bb':A'\rightarrow C$ also belongs to $\R'$.
By Corollary <ref> we see that $hg$ also belongs to $P\R'$.

The class $\R$ is not closed under compositions in general.
By Lemma <ref>, the class $P\R$ is closed under compositions.
Indeed, in the context of Lemma <ref>, suppose that $j:B\rightarrow C$ and $j':C\rightarrow D$ are two objects in $P\R$.
Then the object $v:E\rightarrow A'$ belongs to $\R$ and hence the sequence on the leftmost column is homotopy short exact.
Then the pullback of the cospan $L$ is homotopy bicartesian and hence the sequence in the middle row is homotopy short exact.
Thus the composition $j'j:B\rightarrow D$ belongs to $\R$.

Notice that the homotopy pullback of $j:B\rightarrow C$ along any morphism in $Z^0(\A)$ is in $P\R$.
Thus, the homotopy pullback of $j'j$ along any morphism in $Z^0(\A)$ is a composition of objects in $P(\R)$ and hence is in $\R$ by the above discussion.
Therefore $j'j$ is in $P\R$.
So we have proved
The class $P\R$ is closed under compositions.

Suppose that the class $\R'$ contains $\Id_{A}:A\rightarrow A$ for each $A\in \A$, and is stable under compositions and pullbacks.
If the morphism $A\rightarrow 0$ belongs to $\R'$ for each $A\in \A$, then $Q\R'$ is closed under compositions and satisfies Axiom $\Ex3$: if a composition $A\xrightarrow{a}B\xrightarrow{b}C $ belongs to $Q\R'$ and $b$ admits a homotopy kernel, then $b\in Q\R'$.

The class $P\R$ satisfies the conditions in Proposition <ref>.

Since the cospan of an object in $\R'$ along any morphism in $Z^0(\A)$ admits a homotopy pullback, it is clear that $Q\R'$ is closed under compositions by Corollary <ref>.

We show that $Q\R'$ satisfies Axiom ${\Ex3}$.
Let $g:D\rightarrow E$ and $h:E\rightarrow F$ be two morphisms in $Z^0(\A)$ such that the composition $c=hg$ belongs to $Q\R'$ and that $h$ admits a homotopy kernel

\[
\begin{tikzcd}
K\ar[r,"k"]\ar[rr,bend right=8ex,"u"swap]&E\ar[r,"h"]&F\mathrlap{.}
\end{tikzcd}
\]

Then the object in $\rep(\overline{\Sq},\A)$

\[
\begin{tikzcd}
K\oplus D\ar[r,"{[}0{,}\;1{]}"]\ar[d,"{[}k{,}\;g{]}"swap]\ar[rd,"{[}u{,}\;0{]}"{swap}]&D\ar[d,"c"]\\
E\ar[r,"h"swap] &F
\end{tikzcd}
\]

is homotopy cartesian.
Since the object $K\oplus D\xrightarrow{[0,\;1]}D$ belongs to $\R'$, by definition we have that $h$ belongs to $\R'$.

Let $f:C\rightarrow F$ be arbitrary morphism in $Z^0(\A)$.
Consider the homotopy pullback (<ref>) of $h$ along $f$ and the homotopy pullback of $hg$ along $f$.
Assume $e\in \R'$.
By Corollary <ref>, we infer that $f\in \R'$ since $\R'$ is stable under pullbacks and $hg \in Q\R'$.
Therefore $h\in Q\R'$ and this finishes the proof.

Suppose that the class $\R'$ is stable under compositions and pullbacks. Assume that the morphism $A\rightarrow 0$ belongs to $\R'$ for each $A\in \A$.
Then the class $PQ\R'$ is stable under compositions and pullbacks and satisfies Axiom $\Ex3$, i.e. the class $PQ\R'$ defines a left exact dg structure on $\A$.

By Proposition <ref>, the class $Q\R'$ is stable under compositions.
So by Proposition <ref>, the class $PQ\R'$ is stable under pullbacks and compositions.
We show that $PQ\R'$ satisfies Axiom $\Ex3$.

Let $a:A\rightarrow B$ and $b:B\rightarrow C$ be morphisms in $Z^0(\A)$ such that $b$ admits a homotopy kernel and that the composition $c=ba$ belongs to $PQ\R'$.
By Proposition <ref>, the morphism $b$ belongs to the class $Q\R'$.
In particular, the cospan of $b$ along any morphism in $Z^0(\A)$ admits a homotopy pullback.
Now the claim follows from Corollary <ref> and the fact that $Q\R'$ satisfies Axiom $\Ex3$ which is proved in Proposition <ref>.

The class $P\R$ satisfies the conditions of Proposition <ref>.  So we have

The class $PQP\R$ defines a left exact dg structure on $\A$, which is the unique maximal left exact dg structure on $\A$.

Combining Corollary <ref> and Proposition <ref>, we have

For each additive dg category $\A$, the greatest exact dg structure exists.

The above theorem generalizes Rump's result on the maximal exact structure on an additive category, cf. [125] and also [129, 25].

Let us turn to the special case when $H^0(\A)$ is divisive (=weakly idempotent complete), i.e. each split epimorphism has a kernel.
The following lemma shows that an additive dg category with homotopy kernels is divisive.

Let $\A$ be a dg category.
Let $\overline{p}:Y\rightarrow Z$ be a split epimorphism in $H^0(\A)$. If the object $p:Y\rightarrow Z$ in $\Mor(\A)$ admits a homotopy kernel, then $\overline{p}$ admits a kernel in $H^0(\A)$.

Suppose we have a homotopy left exact sequence

\[
\begin{tikzcd}
X\ar[r,"f"]\ar[rr,bend right,"h"swap]&Y\ar[r,"p"]&Z
\end{tikzcd}
\]

where $d(h)=-pf$.
For each object $A\in\A$, it induces a long exact sequence

\[
\begin{tikzcd}[column sep=tiny]
\ldots\ar[r]&H^{-1}\A(A,X)\ar[r]&H^{-1}\A(A,Y)\ar[r,two heads]&H^{-1}\A(A,Z)\ar[r]&H^0\A(A,X)\ar[r]&H^0\A(A,Y)\ar[r,two heads]&H^0\A(A,Z).
\end{tikzcd}
\]

Since $\overline{p}:Y\rightarrow Z$ is an split epimorphism in $H^0(\A)$, we have that the induced map $H^{-1}\A(A,Y)\rightarrow H^{-1}\A(A,Z)$ is also surjective for each $A\in\A$.
Therefore by the above long exact sequence, we have that $\overline{f}$ is a kernel of $\overline{p}$.

Let $(\A,\mathcal S)$ be a left exact dg category such that $H^0(\A)$ is divisive.
Then there is a universal exact morphism in $\Hqe$

\[
\]

to a pretriangulated dg category. Put $\D^b(\A)=H^0(\D^b_{dg}(\A))$. If $\A$ is connective, then this morphism satisfies the following properties

a) It induces a quasi-fully faithful morphism from $\tau_{\leq 0}\A\rightarrow \tau_{\leq 0}\D^b_{dg}(\A)$.
Let $\A'\subseteq \D^b_{dg}(\A)$ be the essential image.

b) The conflations of $\A$ are inherited from $\D^b_{dg}(\A)$.

c) The inclusion $H^0(\A)\rightarrow \D^b(\A)$ is stable under kernels of retracts.

d) The inclusion $\A'\hookrightarrow \D^b_{dg}(\A)$ is stable under special pullbacks, i.e. if we are given a diagram of the following form

\[
\begin{tikzcd}
\end{tikzcd}
\]

where the objects $X$, $Y$, $Z$ and $W$ are in $\A'$, and the second row is a conflation in $\D^b_{dg}(\A)$, and the right square is a homotopy pullback square in $\D^b_{dg}(\A)$, then $Y'$ is also an object in $\A'$.

By construction $\D^b_{dg}(\A)$ is the dg quotient $\pretr(\A)/\N_{dg}$ where $\N_{dg}$ is the full dg subcategory of $\pretr(\A)$ generated by the totalizations of conflations in $\A$.
The proof of a) is similar to the proof of Theorem <ref>.
The assertion c) follows from the fact that $H^0(\A)$ is divisive.

We show the assertion b).
Consider a triangle in $\D^b(\A)$

\[
\begin{tikzcd}
A\ar[r,"\overline{f}"]&B\ar[r,"\overline{\jmath}"]&C\ar[r,"u/t"]&\Sigma A
\end{tikzcd}
\]

where $A$, $B$ and $C$ are objects in $\A$ and we keep the same notation for the corresponding objects in $\D^b(\A)$.
We proceed as in the proof of Theorem <ref>.
We assume the roof $u/t$ is of the form

\[
\begin{tikzcd}
C&&\Sigma A
\end{tikzcd}
\]

where $N'=\Cone(t)$ is the totalization of a conflation $X'$ of the form

\[
\begin{tikzcd}
A'\ar[r,"f'"]\ar[rr,"h"swap, bend right=8ex]&B'\ar[r,"j'"]&C'.
\end{tikzcd}
\]

Then the morphisms $C\rightarrow N'$ and $\Sigma^{-1}N'\rightarrow \Sigma A$ give rise to morphisms $\overline{c}:C\rightarrow C'$ and $\overline{a}:A'\rightarrow A$ in $H^0(\A)$.
Claim: the morphism $j:B\rightarrow C$ admits a homotopy kernel which is homotopy short exact in $\A$.
Indeed, there is an isomorphism from the above triangle to the following triangle

\[
\begin{tikzcd}
\Sigma^{-1}\Cone(j)\ar[r]&B\ar[r,"\overline{\jmath}"]&C\ar[r,""]&\Cone(j)
\end{tikzcd}
\]

which restricts to identities of $B$ and $C$.
Let $s:A\rightarrow \Sigma^{-1}\Cone(j)$ be the restricted morphism in $\D^b(\A)$.
By the quasi-full faithfulness, this morphism lifts to a morphism in $\pretr(\A)$ and from this morphism we get a homotopy kernel of $j$ which is homotopy short exact. Thus it is enough to show that $j$ is a deflation.

Apply Axiom $\Ex2$ and we get the following diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

where for simplicity we omit the homotopies and where $p$ is a deflation.
Then we have the following diagram in $\D^b(\A)$

\[
\begin{tikzcd}
\end{tikzcd}
\]

By the above claim, the morphism $j:B\rightarrow C$ admits a homotopy kernel in $\A$.
By the quasi-full faithfulness of $F$ and Axiom $\Ex3$, we infer that $j$ is a deflation.

It is direct to verify the assertion d), using Axiom $\Ex2$ and the assertion b).

The converse of Proposition <ref> also holds. Let $\D$ be a pretriangulated dg category. If $\A\subseteq \D$ is a full dg subcategory which is closed under special pullback in the sense of Proposition <ref> d) and which is stable under kernels of retracts, then $\A$ with the conflations inherited from $\D$ is a divisive left exact dg category.

Let $\A$ be an additive dg category such that $H^0(\A)$ is divisive.
Let $\R'$ be a class which is stable under pullbacks.
Let $a:A\rightarrow B$ and $b:B\rightarrow C$ be morphisms in $Z^0(\A)$.
If the composition $c=ba$ belongs to $\R'$, then $b$ admits a homotopy kernel.

The cospan of $c$ along $b$ admits a homotopy pullback $X$

\[
\begin{tikzcd}
\end{tikzcd}
\]

We view $X$ as a morphism from $e$ to $c$ in $H^0(\Mor(\A))$.
By Lemma <ref>, the obvious morphism from $a$ to $c$ factors through the morphism $X$ and hence the morphism $\overline{d}:D\rightarrow A$ is a split epimorphism in $H^0(\A)$.
Since $H^0(\A)$ is divisive, it admits a kernel $K$ and hence the object $d\in H^0(\Mor(\A))$ is isomorphic to the canonical projection $K\oplus A\rightarrow A$ and hence admits a homotopy kernel.
By Corollary <ref>, the object $b$ also admits a homotopy kernel.

Let $\A$ be an additive dg category such that $H^0(\A)$ is divisive.
Then $P\R$ defines the greatest left exact structure on $\A$.

It is clear that the class $P\R$ is stable under compositions and pullbacks.
It remains to check Axiom $\Ex3$ for $P\R$.

Let $a:A\rightarrow B$ and $b:B\rightarrow C$ be morphisms in $Z^0(\A)$ such that the composition $c=ba$ belongs to $P\R$. We claim that $b\in \R$.
Note that $b$ admits a homotopy kernel $X$ by Proposition <ref>

\[
\begin{tikzcd}
K\ar[r,"k"]\ar[rr,"h"swap, bend right=8ex]&B\ar[r,"b"]&C
\end{tikzcd}
\]

We have a morphism of 3-term h-complexes

\[
\begin{tikzcd}
K\ar[rr,"0",bend left=8ex]\ar[d,equal]\ar[r,"v"]\ar[rd,"0"{red,swap},red]\ar[rrd,"0"{blue,swap,near end},blue,bend left=2ex]&K\oplus A\ar[r,"{[}0{,}\;1{]}"]\ar[d,"u"swap]\ar[rd,"{[}-h{,}\;0{]}"{red,near start},red]&A\ar[d,"c"]\\
K\ar[r,"k"swap]\ar[rr,"h"swap, bend right=8ex]&B\ar[r,"b"swap]&C
\end{tikzcd}
\]

where $u=[k,\;a]$, $v=\begin{bmatrix}1\\0\end{bmatrix}$.
It is not hard to see that the right-hand square is homotopy cartesian.
So the morphism $u$ belongs to $\R$.
By Corollary <ref> and Proposition <ref>, the right-hand square is homotopy bicartesian.
By Proposition <ref>, the 3-term h-complex $X$ is homotopy short exact.
Therefore $b$ belongs to $\R$.

It remains to show that $b$ belongs to $P\R$.
Let $f:F\rightarrow C$ be any object in $H^0(\Mor(\A))$.
Consider the following diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

By Corollary <ref>,  the homotopy pullback $c'$ of $c$ along $f$ is homotopic to the composition of some morphism $a'$ with the homotopy pullback $b'$ of $b$ along $f$.
By the above, we have that $b'$ belongs to $\R$.
Thus $b$ belongs to $P\R$.

A morphism $b:B\rightarrow C$ is a semi-stable homotopy cokernel if it belongs to $P\R$.
Dually, one defines the notion of semi-stable homotopy kernel.
A homotopy short exact sequence

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,"h"swap,bend right=8ex]&B\ar[r,"j"]&C
\end{tikzcd}
\]

is stable if $f$ is a semi-stable homotopy kernel and if $g$ is a semi-stable homotopy cokernel.
If the additive dg category $\A$ is such that $H^0(\A)$ is divisive, then by Theorem <ref>, the homotopy short exact sequences in the greatest exact dg structure on $\A$ are exactly the stable ones.

In [129], the authors showed that in pre-abelian categories, i.e. categories with kernels and cokernels, the greatest exact structure consists of the class of all stable kernel-cokernel pairs.
In [25], it was generalized to a more general class of additive categories, namely
the weakly idempotent complete ones.
In [126], Rump gave a counterexample where not every stable short exact sequence in an additive category is in the greatest exact structure.

Let $\C$ be an abelian category having enough projectives and enough injectives.
A full subcategory of $\C$ is resolving if it is stable under extensions, stable under kernels of epimorphisms, stable under taking direct summands and contains all the projectives, cf. <cit.>. Dually one defines the notion of coresolving subcategory.
Let $\E$ be a both resolving and coresolving subcategory of $\C$.
Then it is direct to check that a kernel in $\E$ is a monomorphism in $\C$.
Therefore a kernel-cokernel pair in $\E$ is a short exact sequence in $\C$.
Thus the inherited exact structure on $\E$ from $\C$ is the maximal exact structure on $\E$.
For example, the category $\rep_{\rm{l.f.}}(H)$ of locally free modules over the generalized path algebra $H$ is a both resolving and coresolving subcategory of the ambient abelian category $\rep(H)$, cf. <cit.>, and thus the exact structure of $\rep_{\rm{l.f.}}(H)$ inherited from $\rep(H)$ is the maximal exact structure.

Let $\C$ be an arbitrary abelian category.
If $\E$ is an extension-closed subcategory of $\C$ which is stable under the passage to subobjects, then a kernel-cokernel pair in $\E$ is necessarily a short exact sequence in $\C$. Thus the inherited exact structure on $\E$ from $\C$ is the maximal exact structure. For example, let $Q$ be an acyclic quiver and $\B$ an abelian category. Let $\rep(Q,\B)$ be the category of representations of $Q$ in $\B$, i.e. the category of functors $Q\rightarrow \B$, where $Q$ is considered as a category in the natural way. The monomorphism category, denoted by $\mathrm{mono}(Q,\B)$, is a full subcategory of $\rep(Q,\B)$ which consists of all representations $(M_i,M_{\alpha})_{i\in Q_0, \alpha\in Q_1}$ for which

\[
\begin{tikzcd}
M_{i,in}:\bigoplus_{\alpha\in Q_1,t(\alpha)=i} M_{s(\alpha)}\ar[r,"(M_{\alpha})_{\alpha}"]& M_i
\end{tikzcd}
\]

is a monomorphism for all vertices $i$, where $s(\alpha)$ and $t(\alpha)$ denote the source and target of the arrow $\alpha$.
It is clear that $\mathrm{mono}(Q,\B)$ is stable under extensions and the passage to subobjects in $\rep(Q,\B)$, cf. [42]. Thus the exact structure of $\mathrm{mono}(Q,\B)$ inherited from $\rep(Q,\B)$ is the maximal exact structure.

We recall the setting in Example <ref>.
Let $A$ be a finite-dimensional $k$-algebra.
Let $\mathcal H^{[-1,0]}(\proj A)$ be the full subcategory of $\T=\mathcal H^b(\proj A)$ consisting of two-term complexes $P^{-1} \to P^0$ of finitely generated projective $A$-modules.
Let $\A=\C^{b}_{dg}(\proj A)$ be the canonical enhancement of
${\mathcal H}^b(\proj A)$ and $\A'$ its full dg subcategory on the objects $P^{-1} \to P^0$.
For an $A$-module $M$, we denote by $M^*$ its $A$-dual.

Consider the algebra $A$ given by a quiver with relations as follows

\[
\begin{tikzcd}
&2\ar[rd,""{name=b,swap,near start},""{name=b',swap, near end},"b",""{name=b'',swap}]&\\
1\ar[ru,""{name= a,swap,near end},"a"]&&3\mathrlap{.}\ar[ll,""{name=c, swap,near start},"c"]\arrow[from = a, to = b,bend right=20, dashed, no head]\ar[from=b'',to=c,bend right=20,dashed,no head]
\end{tikzcd}
\]

There exists a homotopy exact squence $X$ in $\A'$ which is not homotopy short exact in $\A$

\[
\begin{tikzcd}
S_2\ar[r]\ar[rr,bend right=8ex,"0"swap]&P_3\ar[r]&P_1
\end{tikzcd}
\]

where $S_2$ stands for the two-term complex $P_1\rightarrow P_2$.
We claim that this homotopy short exact sequence is stable.
As a consequence, the natural exact dg structure on $\A'$ is not the greatest one.

The key point is that the map $P_3\rightarrow P_1$ is a right $\add(P_2\oplus P_3)$-approximation of $P_1$ and the map $P_2^{*}\rightarrow P_1^*$ is a right $\add (P_2^*\oplus P_3^*)$-approximation of $P_1^*$.

Let $T=(T^{-1}\rightarrow T^0)$ be an object in $\A'$.
Consider a morphism $S_2\rightarrow P_3$ in $Z^0(\A')$.
Take the $A$-dual and we get

\[
\begin{tikzcd}
&((T^{0})^*\rightarrow (T^{-1})^*)\ar[d]\\
(P_3^*\rightarrow 0)\ar[r]&(P_2^*\rightarrow P_1^*)\mathrlap{.}
\end{tikzcd}
\]

We have the following diagram

\[
\begin{tikzcd}
0\ar[r]&M\ar[r]\ar[d,two heads]&P_2^*\oplus (T^{-1})^*\ar[r]\ar[d,two heads]&P_1^*\\
\end{tikzcd}
\]

The image of the map $P_2^*\oplus (T^{-1})^*\rightarrow P_1^*$ is either $P_1^*$ so that the second row is split exact, or equals the image of the map $P_2^*\rightarrow P_1^*$ so that $N=(T^{-1})^*$ and the first column is split exact.

In the first case, $M$ is a projective module and $P_1$ is a direct summand of $T^{-1}$.
Assume that $T^{-1}$ decomposes as $P_1\oplus K$.
Then the cospan $L$

\[
\begin{tikzcd}
\end{tikzcd}
\]

admits a homotopy pushout which is homotopy bicartesian.
So we have the following diagram

\[
\begin{tikzcd}
(P_1\oplus K\rightarrow T^0)\ar[r]&(P_2\oplus K\rightarrow T^0\oplus P_3)\ar[r]&P_1
\end{tikzcd}
\]

and it is direct to check the second row is homotopy left exact and hence homotopy exact.

In the second case, we have $M=S_3'\oplus (T^{-1})^*$ and hence the cospan $L$ admits a homotopy pushout which is homotopy bicartesian.
Similarly as above, one checks that the pushout of $X$ along $S_2\rightarrow T$ is homotopy short exact.
Therefore, the morphism $S_2\rightarrow P_3$ is a semi-stable homotopy kernel.
Similarly, the morphism $P_3\rightarrow P_1$ is a semi-stable homotopy cokernel and hence the homotopy short exact sequence $X$ is stable.

§ EXTRIANGULATED CATEGORIES

In this appendix, we give two related constructions of closed subbifunctors, one of them generalizing the construction by Ogawa <cit.> in the case of triangulated categories, and some diagram lemmas in extriangulated categories.

§.§ Closed subbifunctors associated with extension-closed subcategories

Let $\N\subset \C$ be an extension-closed subcategory.
Then we have the following closed subbifunctors of $\mathbb E$

\[
\mathbb E^{\N}: \C^{op}\times \C\rightarrow \Ab,\;\; (C,A)\mapsto\sum_{f:N\rightarrow A, N\in \N}\Im(\mathbb E(C,N)\rightarrow \mathbb E(C,A))
\]

\[
\mathbb E_{\N}: \C^{op}\times \C\rightarrow \Ab,\;\; (C,A)\mapsto\sum_{f:C\rightarrow N, N\in \N}\Im(\mathbb E(N,A)\rightarrow \mathbb E(C,A))
\]

such that $\mathbb E^{\N}|_{\mathbb \N}=\mathbb E|_{\mathbb \N}=\mathbb E_{\N}|_{\mathbb \N}$.
The closed subbifunctor $\mathbb E^{\N}$ (resp. $\mathbb E_{\N}$) is called the subbifunctor left generated (resp. right generated) by $\N$.

When $\C$ is a Quillen exact category, this gives a way to construct Quillen exact substructures of $\C$.

We show the case for $\mathbb F=\mathbb E_{\N}$.
The case for $\mathbb E^{\N}$ follows by duality.
By <cit.>, it is enough to show that $\mathfrak s|_{\mathbb F}$-deflations are closed under compositions.

Since $\N$ is an additive subcategory, it is direct to check that an $\mathfrak s|_{\mathbb F}$-conflation $\delta\in\mathbb F(C,A)$ is in the image of a map

\[
\mathbb E(f,A): \mathbb E(N,A)\rightarrow \mathbb E(C,A)
\]

for some object $N\in \N$ and some map $f:C\rightarrow N$.

Suppose $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ are two $\mathfrak s|_{\mathbb F}$-deflations.
We have the following diagrams made of $\mathbb E$-triangles

\[
\begin{tikzcd}
U\ar[r] &X'\ar[r,"f'"swap]&N\ar[r,"\delta"swap,dashed]&\;
\end{tikzcd}\;,
\]

\[
\begin{tikzcd}
V\ar[r] &Y'\ar[r,"g'"swap]&N'\ar[r,"\mu"swap,dashed]&\;
\end{tikzcd}\;.
\]

By <cit.>, we may choose the maps $j'$ and $k'$ such that we have the following $\mathbb E$-triangles

\[
\begin{tikzcd}
X\ar[r,"\begin{bmatrix}j'\\f\end{bmatrix}"]&X'\oplus Y\ar[r,"{[}f'{,}-j{]}"]&N\ar[r,"e_{*}(\delta)",dashed]&\;
\end{tikzcd},
\]

\[
\begin{tikzcd}
Y\ar[r,"\begin{bmatrix}k'\\g\end{bmatrix}"]&Y'\oplus Z\ar[r,"{[}g'{,}-k{]}"]&N'\ar[r,"d_{*}(\mu)",dashed]&\;
\end{tikzcd}.
\]

So we have the following diagram

\[
\begin{tikzcd}
Y\ar[r,"\begin{bmatrix}k'\\g\end{bmatrix}"]\ar[d,"j"swap]&Y' \oplus Z\ar[d,dashed,red,"{[}u{,}\;v{]}"]\ar[r,"{[}g'{,}-k{]}"]&N'\ar[r,dashed,"d_{*}(\mu)"]\ar[d,equal]&\;\\
N\ar[r,"x"swap]& W\ar[r,"y"swap]&N'\ar[r,dashed,"j_*d_*(\mu)"swap]&\;
\end{tikzcd}.
\]

Since $\N$ is an extension-closed subcategory, we have $W\in\N$.
By Lemma <ref>, there exists a morphism $[u,\;v]:Y'\oplus Z\rightarrow W$ such that the following sequence is an $\mathbb E$-triangle

\[
\begin{tikzcd}
Y\ar[r,"\begin{bmatrix}-j\\k'\\g\end{bmatrix}"]&N\oplus Y'\oplus Z\ar[r,"{[}x{,}\;u{,}\; v{]}"]&W\ar[rr,dashed,"\theta=y^{*}j_{*}d_{*}(\mu)"]&&\;.
\end{tikzcd}
\]

We have the following isomorphism of sequences

\[
\begin{tikzcd}
X\ar[r,"\begin{bmatrix}j'\\f\\k'f \end{bmatrix}"]\ar[d,equal]& X'\oplus Y\oplus Y'\ar[d,"s"]\ar[r,"\begin{bmatrix}f'\;\;-j\;\;0\\0\;\;-k'\;\;1\end{bmatrix}"]&N\oplus Y'\ar[d,equal]&\\
X\ar[r,"\begin{bmatrix}j'\\f\\0 \end{bmatrix}"swap]&X'\oplus Y\oplus Y'\ar[r,"\begin{bmatrix}f'\;\;-j\;\;0\\0\;\;0\;\;1\end{bmatrix}"swap]&N\oplus Y'\ar[r,dashed,"e_*(\delta)\oplus 0"]&\;
\end{tikzcd}
\]

where $s=\begin{bmatrix}1&0&0\\0&1&0\\0&-k'&1\end{bmatrix}$.
Thus the first row is also a conflation.

Now we apply the pasting law for homotopy cartesian squares (cf. [60]) to the following diagram

\[
\begin{tikzcd}
X'\oplus Y'\ar[r,"\begin{bmatrix}f'\;\;0\\0\;\;1\end{bmatrix}"swap]&N\oplus Y'\ar[r,"{[}-x{,}\; u{]}"swap]&W
\end{tikzcd}
\]

and by Lemma <ref>, the morphism $[-x,\;u]$ is an $\mathfrak s$-deflation.
We see that $gf$ is an $\mathfrak s|_{\mathbb F}$-deflation.

Let $(\C,\mathbb E,\mathfrak s)$ be an extriangulated category and $\N$ an extension-closed subcategory.
Then we have the following closed subbifunctors of $\mathbb E$

\[
\mathbb E^{L}_{\N}:\C^{op}\times \C\to \mathrm{Ab},\quad (C,A)\mapsto \{\delta\in \mathbb E(C,A)| \text{for any $N\in \N$ and $s:N\rightarrow C$, $s^{*}(\delta)\in \mathbb E^{\N}(N,A)$})\}
\]

\[
\mathbb E^{R}_{\N}:\C^{op}\times \C\to \mathrm{Ab},\quad (C,A)\mapsto \{\delta\in \mathbb E(C,A)| \text{for any $N\in\N$ and $t:A\rightarrow N$, $t_{*}(\delta)\in \mathbb E_{\N}(C,N)$})\}
\]

where $\mathbb E^{\N}$ and $\mathbb E_{\N}$ are the closed subbifunctors defined in Lemma <ref>.

We prove that $\mathbb E^{L}_{\N}$ is a closed subbifunctor. The other case is similar.

Put $\mathbb E'=\mathbb E^{\mathcal N}$ and $\mathbb F=\mathbb E^{L}_{\N}$.

\[
\begin{tikzcd}
\end{tikzcd},
\]

\[
\begin{tikzcd}
\end{tikzcd}
\]

be $\mathfrak s|_{\mathbb F}\mbox{-}$triangles.
Let $N$ be an object in $\N$ and $s:N\rightarrow C$ any morphism in $\C$.
Then we have the following diagram in $\C$

\[
\begin{tikzcd}
\end{tikzcd}
\]

where the first row is an $\mathfrak s\mbox{-}$triangle.
By Axiom $(\mathrm{ET3})^{op}$, there exists a morphism $(a,\Id_{C}):\theta\rightarrow \delta$ which is realized by $(a,p,\Id_{C})$ and in particular, we have $a_*(\theta)=\delta$.

We have the following diagram in $\C$ made of $\mathfrak s\mbox{-}$triangles

\[
\begin{tikzcd}
\end{tikzcd}.
\]

By Lemma <ref>, there exists a morphism $z:F\rightarrow B$ such that

\begin{equation}\label{dia:Etriangle}
\begin{tikzcd}
F\ar[r,"\begin{bmatrix}z\\q\end{bmatrix}"]&B\oplus N \ar[r,"{[}jp{,}\;-s{]}"]&C\ar[r,dashed,"u_{*}(\theta)"]&\;
\end{tikzcd}
\end{equation}

is an $\mathbb E\mbox{-}$triangle.

By definition, we have $s^*(\delta)\in\mathbb E'(N,E)$ and hence we have some morphism $t:N' \rightarrow E$ and an $\mathbb E$-extension $\mu\in \mathbb E(N,N')$ with $N'\in \mathcal N$ and $t_*(\mu)=s^*(\delta)$. So we have the following diagram

\[
\begin{tikzcd}
\end{tikzcd}.
\]

By definition, we have $(nm)^{*}(\psi)\in \mathbb E'(N'',H)$ and hence we have some morphism $g:\tilde{N}\rightarrow H$ and an $\mathbb E$-extension $\gamma\in \mathbb E(N'', \tilde{N})$ with $\tilde{N}\in \N$ and $g_*(\gamma)=(nm)^{*}(\psi)$.
So we have the following diagram

\[
\begin{tikzcd}
\tilde{N}\ar[r,"v"]\ar[d,"g"]&\tilde{N'}\ar[d,"k"]\ar[r,"w"]&N''\ar[d,equal]\ar[r,"\gamma",dashed]&\;\\
\end{tikzcd}.
\]

So we have the following diagram

\[
\begin{tikzcd}
\tilde{N'}\ar[r, "w"]\ar[rd,dashed,"c"description]\ar[ddr,bend right=6ex,"lk"swap]&N''\ar[rd,"b"]&\\
\end{tikzcd}
\]

By the $\mathbb E$-triangle <ref>, there exists a (non-unique) morphism $c:\tilde{N'}\rightarrow F$ making the above diagram commute.
We apply axiom $(\mathrm{ET3})^{op}$ to the following diagram

\[
\begin{tikzcd}
\tilde{N}\ar[r,"v"]\ar[d,dashed,"d"swap]&\tilde{N'}\ar[d,"c"]\ar[r,"w"]&N''\ar[d,"b"]\ar[r,"\gamma",dashed]&\;\\
\end{tikzcd}
\]

and obtain a morphism of $\mathbb E$-triangles $(d,b):\gamma\rightarrow s^*(\theta)$.
This implies that $b^*(s^*(\theta))$ is an $\mathbb E'$-extension.
Since $b$ is an $\mathbb E'$-deflation, by <cit.> we have $s^*(\theta)\in \mathbb E'(N,A)$ and hence $jp:B\rightarrow C$ is an $\mathbb F$-deflation. This finishes the proof.

§.§ Some diagram lemmas

Let $(\mathcal C,\mathbb E,\mathfrak s)$ be an extriangulated category.
The following lemma is called (ET4-2) in [92].

\[
\begin{tikzcd}
\end{tikzcd}
\]

be any $\mathbb E\mbox{-}$triangle, let $f:A\rightarrow D$ any morphism, and let

\[
\begin{tikzcd}
\end{tikzcd}
\]

be any $\mathbb E\mbox{-}$triangle realizing $f_{*}\delta$.
Then there is a morphism $g$ which gives a morphism of $\mathbb E\mbox{-}$triangles

\[
\begin{tikzcd}
A\ar[r,"x"]\ar[d,"f"swap]&B\ar[r,"y"]\ar[d,"g"red,dashed]&C\ar[d,equal]\ar[r,dashed,"\delta"]&\ \\
D\ar[r,"d"swap]            &E\ar[r,"e"swap]              &C\ar[r,dashed,"f_{*}\delta"swap]           &\
\end{tikzcd}
\]

and moreover, the sequence

\[
\begin{tikzcd}
A\ar[r,"\begin{bmatrix} {-}f \\ x \end{bmatrix}"] & D\oplus B\ar[r,"{[}d{,}\; g{]}"]&E\ar[r,dashed,"e^{*}\delta"]&\
\end{tikzcd}
\]

becomes an $\mathbb E\mbox{-}$triangle.

A commutative square

\begin{equation}\label{dia:extrihomotopycartesian}
\begin{tikzcd}
\end{tikzcd}
\end{equation}

is a homotopy cartesian square if there exists an $\mathbb E$-triangle

\[
\begin{tikzcd}
A\ar[r,"\begin{bmatrix}f\\a\end{bmatrix}"]&B\oplus C \ar[r,"{[}-b{,}\; u{]}"]&D\ar[r,dashed,"\delta"]&\;
\end{tikzcd}
\]

in $\C$. The $\mathbb E$-extension $\delta$ is called a differential for the square.

Consider a homotopy cartesian square <ref> with a differential $\delta$.
Then $f$ is an $\mathfrak s$-deflation (resp. $\mathfrak s$-inflation) if and only if so is $u$.

We only prove the deflation part of the statement.
Suppose that $f$ is an $\mathfrak s$-deflation. Let

\[
\begin{tikzcd}
\end{tikzcd}
\]

be an $\mathbb E$-triangle.
We apply <cit.> to the following diagram

\[
\begin{tikzcd}
C\ar[r]&B\oplus C\ar[d,"{[}-b{,}\; u{]}"]\ar[r,"{[}1{,}\;0{]}"]&B\ar[r,"0",dashed]&\;\\
\end{tikzcd}
\]

and hence $u:C\rightarrow D$ is also an $\mathfrak s$-deflation.

Conversely, suppose that $u$ is an $\mathfrak s$-deflation.

\[
\begin{tikzcd}
\end{tikzcd}
\]

be an $\mathbb E$-triangle.
We have the following diagram

\[
\begin{tikzcd}
E\ar[r,"g'"]\ar[d,equal]&A'\ar[r,"f'"]\ar[d,"a'"red,dashed]&B\ar[d,"b"l]\ar[r,dashed,"b^{*}(\mu)"]&\ \\
E\ar[r,"h"swap]            &C\ar[r,"u"swap]              &D\ar[r,dashed,"\mu"swap]           &\
\end{tikzcd}
\]

By the dual of Lemma <ref>, there exists a morphism $a':A'\rightarrow C$ making the above diagram commute and such that the right hand side square is homotopy cartesian with a differential given by $g'_{*}(\mu)$.
It follows that there exists an isomorphism $j:A\rightarrow A'$ such that $f'j=f$ and hence $f$ is also an $\mathfrak s$-deflation.

The following lemma is called (ET4-5) in [92].

\[
\begin{tikzcd}
\end{tikzcd}
\]

\[
\begin{tikzcd}
\end{tikzcd}
\]

be $\mathbb E-$triangles and $g:B_1\rightarrow B_2$ be a morphism such that $y_2g=y_1$.
Then there is a morphism $f:A_1\rightarrow A_2$ which gives a morphism of $\mathbb E\mbox{-}$triangles

\[
\begin{tikzcd}
A_1\ar[r,"x_1"]\ar[d,"f"swap,dashed]&B_1\ar[r,"y_1"]\ar[d,"g"]&C\ar[d,equal]\ar[r,dashed,"\delta"]&\ \\
A_2\ar[r,"x_2"swap]                         &B_2\ar[r,"y_2"swap]              &C\ar[r,dashed,"\delta'"swap]                  &\
\end{tikzcd}
\]

and moreover, the squence

\[
\begin{tikzcd}
A_1\ar[r,"\begin{bmatrix} {-}f \\ x_1 \end{bmatrix}"] & A_2\oplus B_1\ar[r,"{[}x_2{,}g{]}"]&B_2\ar[r,dashed,"y_2^{*}\delta"]&\
\end{tikzcd}
\]

becomes an $\mathbb E\mbox{-}$triangle.

Using the above lemma, we prove a slightly stronger version of <cit.>.

Suppose we are given $\mathbb E\mbox{-}$triangles

\[
\begin{tikzcd}
\end{tikzcd}
\]

\[
\begin{tikzcd}
\end{tikzcd}
\]

\[
\begin{tikzcd}
\end{tikzcd}
\]

satisfying $y_2\circ g=y_1$. Then there is an $\mathbb E\mbox{-}$triangle

\[
\begin{tikzcd}
\end{tikzcd}
\]

which makes the following diagram commutative in $\mathcal C$

\[
\begin{tikzcd}
A_1\ar[r,"x_1"]\ar[d,"f"swap,dashed]&B_1\ar[r,"y_1"]\ar[d,"g"]                    &C\ar[d,equal]\\
A_2\ar[r,"x_2"]\ar[d,"k"swap,dashed]                         &B_2\ar[r,"y_2"] \ar[d,"h"]             &C\\
\end{tikzcd}
\]

and satisfy the following equalities:

(i) $f_{*}(\delta)=\delta'$,

(ii) $(x_1)_*(\theta)=\delta''$,

(iii) $h^{*}(\theta)+y_2^{*}(\delta)=0$

and moreover, the sequence

\[
\begin{tikzcd}
A_1\ar[r,"\begin{bmatrix} {-}f \\ x_1 \end{bmatrix}"] & A_2\oplus B_1\ar[r,"{[}x_2{,}g{]}"]&B_2\ar[r,dashed,"y_2^{*}\delta"]&\
\end{tikzcd}
\]

becomes an $\mathbb E\mbox{-}$triangle.

By the previous lemma, there exists a morphism $f:A_1\rightarrow A_2$ which gives a morphism of $\mathbb E-$triangles

\[
\begin{tikzcd}
A_1\ar[r,"x_1"]\ar[d,"f"swap,dashed]&B_1\ar[r,"y_1"]\ar[d,"g"]&C\ar[d,equal]\ar[r,dashed,"\delta"]&\ \\
A_2\ar[r,"x_2"swap]                         &B_2\ar[r,"y_2"swap]              &C\ar[r,dashed,"\delta'"swap]                  &\
\end{tikzcd}
\]

and moreover, the sequence

\[
\begin{tikzcd}
A_1\ar[r,"\begin{bmatrix} {-}f \\ x_1 \end{bmatrix}"] & A_2\oplus B_1\ar[r,"{[}x_2{,}g{]}"]&B_2\ar[r,dashed,"y_2^{*}\delta"]&\
\end{tikzcd}
\]

becomes an $\mathbb E\mbox{-}$triangle.

Thus we have the following diagram in $\mathcal C$

\[
\begin{tikzcd}
&B_1\ar[r,equal]\ar[d,"\begin{bmatrix}0\\1\end{bmatrix}"swap]         &B_1\ar[d,"g"]\\
A_1\ar[d,equal]\ar[r,"\begin{bmatrix} {-}f \\ x_1 \end{bmatrix}"] & A_2\oplus B_1\ar[r,"{[}x_2{,}g{]}"]\ar[d,"{[}1{,}0{]}"swap]     &B_2\ar[d,"h"]\\
\end{tikzcd}
\]

By the dual of <cit.>, there is an $\mathbb E-$triangle

\[
\begin{tikzcd}
\end{tikzcd}
\]

which makes the above diagram commutative in $\mathcal C$ and $h^*(-\theta)=y_2^{*}\delta$ and

\[
\begin{bmatrix}-f\\x_1\end{bmatrix}_{*}(-\theta)+\begin{bmatrix}0\\1\end{bmatrix}_{*}(\delta'')=0.
\]

Applying the morphism $[0,1]_{*}$ to the last equality, we see that $(x_1)_{*}(\theta)=\delta''$.

Similarly, we can prove a slightly stronger version of <cit.>.

Suppose we are given $\mathbb E\mbox{-}$triangles

\[
\begin{tikzcd}
\end{tikzcd}
\]

\[
\begin{tikzcd}
\end{tikzcd}
\]

Then there is a commutative diagram in $\mathcal C$:

\[
\begin{tikzcd}
&A_2\ar[r,equal]\ar[d,"m_2"]          &A_2\ar[d,"x_2"]\\
A_1\ar[r,"m_1"]\ar[d,equal]     &M\ar[r,"e_1"]\ar[d,"e_2"]              &B_2\ar[d,"y_2"]\\
A_1\ar[r,"x_1"swap]                        &B_1\ar[r,"y_1"swap]                             &C
\end{tikzcd}
\]

which satisfies

\[
\mathfrak s(y_2^{*}\delta)=[A_1\xrightarrow{m_1} M\xrightarrow{e_1} B_2],
\]

\[
\mathfrak s(y_1^{*}\delta')=[A_2\xrightarrow{m_2} M\xrightarrow{e_2} B_1],
\]

\[
\]

and moreover, the sequence

\[
\begin{tikzcd}
M\ar[r,"\begin{bmatrix}{e_1}\\e_2\end{bmatrix}"]&B_1\oplus B_2\ar[r,"{[}y_1{,}-y_2{]}"]&C\ar[r,dashed,"(m_1)_*(\delta)"]&\,
\end{tikzcd}
\]

is an $\mathbb E\mbox{-}$triangle.

We apply the dual of Lemma <ref> to the diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

Then there exists a morphism $e_2:M\rightarrow B_1$ such that the above diagram commutes in $\mathcal C$ and

\[
\begin{tikzcd}
M\ar[r,"\begin{bmatrix}{e_2}\\e_1\end{bmatrix}"]&B_1\oplus B_2\ar[r,"{[}y_1{,}-y_2{]}"]&C\ar[r,dashed,"(m_1)_*(\delta)"]&\,
\end{tikzcd}
\]

is an $\mathbb E\mbox{-}$triangle.

We then apply <cit.> to the diagram

\[
\begin{tikzcd}
A_2\ar[r,"x_2"]\ar[d,"m_2",dashed]&B_2\ar[r,"-y_2"]\ar[d,"\begin{bmatrix}0\\1\end{bmatrix}"]                    &C\ar[d,equal]\\
M\ar[r,"\begin{bmatrix}e_2\\e_1\end{bmatrix}"]\ar[d,"k",dashed]                         &B_1\oplus B_2\ar[r,"{[}y_1{,}-y_2{]}"] \ar[d,"{[}1{,}0{]}"]             &C\\
\end{tikzcd}.
\]

So we have an $\mathbb E\mbox{-}$triangle

\[
\begin{tikzcd}
\end{tikzcd}
\]

such that the above diagram commutes (so we have $k=e_2$) and

(i) $(m_2)_{*}(-\delta')=(m_1)_*(\delta)$,

(ii) $(x_2)_{*}(\theta)=0$,

(iii) $[1,0]^{*}(\theta)+[y_1,-y_2]^{*}(-\delta')=0$.

Applying the morphism $\begin{bmatrix}1\\0\end{bmatrix}^{*}$ to the last equality, we get $\theta=y_1^*(\delta')$.

We also have the following slightly stronger version of <cit.>, which is called (ET4-3) in [92].

\[
\begin{tikzcd}
\end{tikzcd}
\]

\[
\begin{tikzcd}
\end{tikzcd}
\]

\[
\begin{tikzcd}
\end{tikzcd}
\]

be $\mathbb E\mbox{-}$triangles satisfying $h=gf$. Then there are morphisms $d_0:D\rightarrow E_0$ and $e_0:E_0\rightarrow F$ such that the following

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[d,equal]                    &B\ar[d,"g"]\ar[r,"f'"]                                        &D\ar[r,"\delta_{f}"]\ar[d,"d_0"]            &\,\\
A\ar[r,"h"]                   &C\ar[d,"g'"]\ar[r,"h_0"]                                   &E_0\ar[r,"\delta_{h}"] \ar[d,"e_0"]          &\,\\
\,                               &F\ar[d,dashed,"\delta_{g}"]\ar[r,equal]          &F\ar[d,dashed,"f'_{*}(\delta_g)"]  &\,\\
\,                                &\,                                                               &\,                                                   &\,
\end{tikzcd}
\]

is a commutative diagram whose third column is an $\mathbb E\mbox{-}$triangle, moreover, $d_0^*(\delta_h)=\delta_f$ and $f_*\delta_h=e_0^*(\delta_g)$ and the sequence

\[
\begin{tikzcd}
B\ar[r,"\begin{bmatrix}g\\f{'}\end{bmatrix}"]           &   C\oplus D\ar[r,"{[}h_0{,}-d_0{]}"]             &E_0\ar[r,dashed,"f_{*}(\delta_h)"]&\,
\end{tikzcd}
\]

is an $\mathbb E\mbox{-}$triangle.

By the dual of Lemma <ref>, there exists a morphism $d_0:D\rightarrow E_0$ such that

\[
\begin{tikzcd}
B\ar[r,"\begin{bmatrix}g\\f'\end{bmatrix}"]&C\oplus D\ar[r,"{[}h_0{,}-d_0{]}"]&E_0\ar[r,dashed,"\theta=f_*(\delta_h)"]&\,
\end{tikzcd}
\]

is an $\mathbb E\mbox{-}$triangle and $(\Id_{A},g,d_0)$ is a morphism of $\mathbb E\mbox{-}$triangles.
In particular, $d_0^{*}(\delta_h)=\delta_f$.

Consider the following commutative diagram

\[
\begin{tikzcd}
B\ar[r,"\begin{bmatrix}g\\f'\end{bmatrix}"]\ar[d,equal]&C\oplus D\ar[r,"{[}h_0{,}-d_0{]}"]\ar[d,"{[}1{,}0{]}"swap]&E_0\ar[r,dashed,"\theta"]\ar[d,dashed,"e_0"]&\,\\
B\ar[r,"g"swap]&C\ar[r,"g'"swap] \ar[d,dashed,"0"swap]&F\ar[d,dashed, "\mu"]\ar[r,dashed,"\delta_g"swap]&\,\\
\,&\,&\,
\end{tikzcd}
\]

By the dual of <cit.>, there exists an $\mathbb E\mbox{-}$triangle

\[
\begin{tikzcd}
\end{tikzcd}
\]

which makes the above diagram commute (hence $d_0=\tilde{d_0}$) and $\begin{bmatrix}0\\-1\end{bmatrix}_{*}(\mu)+\begin{bmatrix}g\\f'\end{bmatrix}_{*}(\delta_g)=0$, $e_0^{*}(\delta_g)=\theta$.

Thus we have $\mu=f'_{*}(\delta_g)$ and $e_0^{*}(\delta_g)=f_*(\delta_{h})$.

The triangulated case for Lemma <ref> is proved in <cit.>.

A contravariant connected sequence of functors is a pair $(T,\epsilon)$ where $T=(T^i)_{i\geq 0}$ is sequence of right $\C$-modules and $\epsilon$ is a collection of morphisms $\epsilon_{\delta}^i:T^i(A)\rightarrow T^{i+1}(C)$ for each $\mathbb E\mbox{-}$extension $\delta\in\mathbb E(C,A)$ and $i\geq 0$ which is natural with respect to morphisms of $\mathbb E$-extensions.
It is a right $\delta$-functor if for any $\mathfrak s\mbox{-}$triangle

\[
\begin{tikzcd}
A\ar[r,"k"]&B\ar[r,"p"] &C\ar[r,dashed,"\delta"]&\;
\end{tikzcd},
\]

the associated sequence (which is a complex by the naturality of $\epsilon$, cf. <cit.>)

\[
\ldots \rightarrow T^{n}(C)\xrightarrow{T^n(p)} T^n(B)\xrightarrow{T^n(k)}T^n(A)\xrightarrow{\epsilon^{n}_{\delta}} T^{n+1}(C)\rightarrow \ldots
\]

is exact.
A morphism $(T,\epsilon)\rightarrow (\tilde{T},\tilde{\epsilon})$ of right $\delta$-functors is a family of morphisms of right $\C$-modules $\theta^i:T^i\rightarrow \tilde{T}^i$ which is compatible with the connecting morphisms, i.e. for each $\delta\in \mathbb E(C,A)$ and each $i\geq 0$, the following diagram is commutative

\[
\begin{tikzcd}
\tilde{T}^i(A)\ar[r,"\tilde{\epsilon}^i_{\delta}"swap]&\tilde{T}^{i+1}(C)
\end{tikzcd}.
\]

Dually one defines the notion of covariant connected sequence of functors and left $\delta$-functors.

We have the following extriangulated analogue of <cit.>.

Let $(T,\epsilon)$ and $(R,\eta)$ be right $\delta$-functors such that $R^i$ is weakly effaceable for each $i>0$.
Then each morphism $R^0\rightarrow T^0$ extends uniquely to a morphism of right $\delta$-functors $(R,\eta)\rightarrow (T,\epsilon)$.

Suppose we have constructed natural morphisms $\theta^j:R^j\iso T^j$ which are compatible with the connecting morphisms for $0\leq j<i$.

Let $C$ be an object in $\C$ and $x\in R^i(C)$ an arbitrary element.
Since $R^i$ is effaceable, there exists a $\mathbb E$-triangle

\[
\begin{tikzcd}
A\ar[r,"k"]&B\ar[r,"p"] &C\ar[r,dashed,"\delta"]&\;
\end{tikzcd},
\]

such that $R^i(p)(x)=0$.
We have the following diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

By the exactness of the top horizontal sequence, there exists an element $y\in R^{i-1}(A)$ such that $\eta_{\delta}^{i-1}(y)=x$.
Since the left square commutes by the construction of $\theta^{i-1}$, it is clear that the element $\epsilon_{\delta}^{i-1}\circ \theta_{A}^{i-1}(y)$ is independent of the choice of $y$.

\[
\begin{tikzcd}
A'\ar[r,"l"]&B'\ar[r,"q"] &C\ar[r,dashed,"\delta'"]&\;
\end{tikzcd}
\]

be another such $\mathbb E$-triangle.
By Lemma <ref>, we have the following diagram

\[
\begin{tikzcd}
A\ar[r,"k"swap]&B\ar[r,"p"swap] &C\ar[r,dashed,"\delta"swap]&\;
\end{tikzcd}
\]

where $m_*(\delta)=-f_*(\delta')$ and the sequence

\[
\begin{tikzcd}
E\ar[r,"\begin{bmatrix}r\\g\end{bmatrix}"]&B'\oplus B\ar[r,"{[}-q{,}\;p{]}"] &C\ar[r,dashed,"m_{*}(\delta)"]&\;
\end{tikzcd}
\]

is an $\mathbb E$-triangle. So we have the following diagram made of $\mathbb E$-triangles

\[
\begin{tikzcd}
A\ar[r,"k"]\ar[d,"m"swap]&B\ar[r,"p"] \ar[d,"\begin{bmatrix}1\\0\end{bmatrix}"swap]&C\ar[r,dashed,"\delta"]\ar[d,equal]&\;\\
E\ar[r,"\begin{bmatrix}g\\r\end{bmatrix}"swap]&B\oplus B'\ar[r,"{[}-q{,}\;p{]}"swap] &C\ar[r,dashed,"m_{*}(\delta)"swap]\ar[d,equal]&\;\\
A'\ar[r,"l"swap]\ar[u,"-f"]&B'\ar[r,"q"swap] \ar[u,"\begin{bmatrix}0\\-1\end{bmatrix}"]&C\ar[r,dashed,"\delta'"swap]&\;
\end{tikzcd}
\]

This implies that $\epsilon_{\delta}^{i-1}\circ \theta_{A}^{i-1}(y)$ is independent of the choice of the $\mathbb E$-triangle.
Similarly as above, one shows that the map $\theta^{i}_{C}: R^{i}(C)\rightarrow T^{i}(C)$ is a morphism of abelian groups.
It is direct to verify that $\theta^i:R^i\rightarrow T^{i}$ is a natural transformation
and it follows directly from the construction that $\theta^i$ is compatible with the connecting morphisms.
By induction on $i$, we have natural transformations $\theta^i:R^i\rightarrow T^i$ for $i\geq 0$ which are compatible with the connecting morphisms. The uniqueness is also clear from the construction of the $\theta^i$.

A $\delta$-functor is a triple $(T,\epsilon,\eta)$ where $T=(T^i)_{i\geq 0}$ be a sequence of $\C$-$\C$-bimodules
and $\epsilon$ and $\eta$ are collections of morphisms $\epsilon_{\delta}^i:T^i(A,-)\rightarrow T^{i+1}(C,-)$ and $\eta_{\delta}^i:T^i(?,C)\rightarrow T^{i+1}(?,A)$ for each $\mathbb E$-extension $\delta\in\mathbb E(C,A)$ and $i\geq 0$ which are natural with respect to morphisms of $\mathbb E$-extensions, and such that for each $\mathfrak s$-triangle

\[
\begin{tikzcd}
A\ar[r,"k"]&B\ar[r,"p"] &C\ar[r,dashed,"\delta"]&\;
\end{tikzcd},
\]

the associated complexes

\[
\ldots \rightarrow T^{n}(C,-)\xrightarrow{T^n(p,-)} T^n(B,-)\xrightarrow{T^n(k,-)}T^n(A,-)\xrightarrow{\epsilon^{n}_{\delta}} T^{n+1}(C,-)\rightarrow \ldots
\]

\[
\ldots \rightarrow T^{n}(?,A)\xrightarrow{T^n(?,k)} T^n(?,B)\xrightarrow{T^n(?,p)}T^n(?,C)\xrightarrow{\eta^{n}_{\delta}} T^{n+1}(?,A)\rightarrow \ldots
\]

are exact in $\Mod \C$, resp. in $\C\Mod$.
A morphism $(T,\epsilon,\eta)\rightarrow (\tilde{T},\tilde{\epsilon},\tilde{\eta})$ of $\delta$-functors is a family of morphisms of $\C$-$\C$-bimodules $\theta^i:T^i\rightarrow \tilde{T}^i$ which is compatible with the connecting morphisms.

Let $(T,\epsilon,\eta)$ a $\delta$-functor.
We assume that

\begin{equation}\label{assumption:delta1}\tag{A.1}
\text{$T^0=\Hom(?,-)$ and $T^i$ are effaceable for $i>0$.}
\end{equation}

By Proposition <ref>, we have that $T^i$ is isomorphic to $\mathbb E^i$ for $i>0$ which is compatible with the connecting morphisms $\epsilon$.
We see that $\epsilon_{\delta}^0:\Hom(A,-)\rightarrow \mathbb E(C,-)$ is given by

\[
(\epsilon_{\delta}^0)_U:\Hom(A,U)\rightarrow \mathbb E(C,U),\;\; f\mapsto f_{*}\delta
\]

for each object $U\in\C$.

\begin{equation}\label{assumption:delta2}\tag{A.2}
\text{We assume that a similar description holds for $\eta_{\delta}^0$.}
\end{equation}

For each $i\geq 2$ and any two $\mathfrak s$-triangles

\[
\begin{tikzcd}
A\ar[r,"k"]&B\ar[r,"p"] &C\ar[r,dashed,"\delta"]&\;
\end{tikzcd},
\]

\[
\begin{tikzcd}
F\ar[r,"l"]&G\ar[r,"q"] &H\ar[r,dashed,"\psi"]&\;
\end{tikzcd}
\]

\begin{equation}\label{assumption:delta3}\tag{A.3}
\text{we assume the following diagram~\ref{dia:bivariantdeltafunctor} is commutative}
\end{equation}

\begin{equation}\label{dia:bivariantdeltafunctor}
\begin{tikzcd}
\end{tikzcd}\;.
\end{equation}

Put $\mathbb E^0=\Hom_{\C}(?,-)$.
From the definition of $\mathbb E^n$ and the construction of the connecting morphisms, cf. <cit.>, it is clear that $(\mathbb E^n)_{n\geq 0}$ together with its connecting morphisms satisfy the above assumptions <ref>–<ref>.

Keep the notations as above.
Let $(R,\gamma,\mu)$ be another $\delta$-functor which satisfies the above assumptions A.1–A.3.
Then there is a unique isomorphism of $\delta$-functors $\theta^i: R^i\rightarrow T^i$
for $i\geq 0$ such that $\theta^0=\Id_{\Hom_{\C}(?,-)}$.

By Proposition <ref>, we have that for each $i\geq 0$, the right $\C$-module $R^i(?,A)$ is naturally isomorphic to $T^i(?,A)$ in $\Mod \C$ and the isomorphism is natural in the variable $A$ and compatible with the connecting morphisms.
Hence we have an isomorphism of $\C$-$\C$-bimodules $\theta^i: R^i\rightarrow T^i$ for each $i\geq 0$.
Similarly $R^i(C,-)$ is naturally isomorphic to $T^i(C,-)$ and this also gives an isomorphism of $\C$-$\C$-bimodules $\lambda^i:R^i\rightarrow T^i$ for each $i\geq 0$.

We identify both $R^1$ and $T^1$ with $\mathbb E$ and by assumption the natural isomorphisms $\theta^1$ and $\lambda^1$ are both given by the identity.
We only need to check that the natural isomorphisms $\theta^i$ and $\lambda^i$ coincide with each other for each $i\geq 2$ and we proceed by induction on $i$.

Let $C$ and $F$ be objects in $\C$ and $x\in R^{i}(C,F)$ for $i\geq 1$.
Since $R^i$ and $R^{i-1}$ are both effaceable, there exists $\mathfrak s$-triangles

\[
\begin{tikzcd}
A\ar[r,"k"]&B\ar[r,"p"] &C\ar[r,dashed,"\delta"]&\;
\end{tikzcd},
\]

\[
\begin{tikzcd}
F\ar[r,"l"]&G\ar[r,"q"] &H\ar[r,dashed,"\psi"]&\;
\end{tikzcd}
\]

such that there exists an element $y\in R^{i-2}(A,H)$ with $x=\mu_{\psi}^{i-1}\gamma_{\delta}^{i-2}(y)=\gamma_{\delta}^{i-1}\mu_{\psi}^{i-2}(y)$ where the second equality is given by the commutativity of the diagram <ref> for the bimodules $R^{i}$, $R^{i-1}$ and $R^{i-2}$.
Since by induction $\theta^{j}$ coincides with $\lambda^{j}$ for $j=i-2$ and $i-1$, we obtain that the isomorphisms $\theta^{i}$ and $\lambda^i$ coincide with each other and this finishes the proof.

$'$ $'$

$'$ $'$

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