Every cyclic, $\dagger$-$*$-autonomous category is a conjugative $*$-autonomous category.

Let $\X$ be a cyclic, $\dagger$-$*$-autonomous category then composing adjoints gives the equivalence  $(\_)^{\dagger^*} \dashv (\_)^{*^\dagger}$. To build a conjugation, however,
we need an equivalence between the same functors: to obtain such an equivalence we use the natural equivalence $\omega: (\_)^{*\dagger} \to (\_)^{\dagger*}$ from the cyclor preserving condition for Frobenius linear functors.  A conjugative equivalence, in addition, requires that the unit and counit of the equivalence be inverses of each of other.
The unit and counit of the equivalence are given by (a) and (b) respectively;

\[ \mbox{\em (a)}~~~~~~~~~  \xymatrix{
\X^{\sf rev} \ar@{=}[rrrr] \ar[dr]_{(\_)^{*^\op}} &
\ar@{}[d]|{\Downarrow ~ \eta_\ox'}&
\X^{\sf rev} \ar@{<-}[ld]^{(\_)^{*^\op}}
& \\
\X^{\op} \ar@{=}[rr] \ar[dr]_{\dagger} &
\ar@{}[d]|{\Downarrow ~ \iota^{-1}} &
\X^{\sf op} \ar@{<-}[ld]^{\dagger^\op} \ar@{}[rr]_{\omega}  \ar@{}[rr]^{\Longrightarrow}  &
\X^{\sf oprev} \ar@{<-}@/^1pc/[llld]^{(\_)^{*^{\sf oprev}}} \ar@/_1pc/[ul]_{\dagger^\rev}  \\
& & \X & &
\]

\[
\mbox{\em (b)}~~~~~~~~~~~~~~ \xymatrix{
& & &
\X^{\rev}  \ar@{<-}@/_1pc/[llld]_{\dagger^\rev} \ar@{<-}[dl]_{(\_)^{*^{\rev}}} \ar[dr]^{(\_)^{*^{\sf op}}} \ar@{}[d]|{\Downarrow ~ \epsilon'_\ox} & &\\
\X^{\sf op rev} \ar@{<-}[rd]_{(\_)^{*^{\sf oprev}}} \ar@{}[rr]_{\omega^{-1}}  \ar@{}[rr]^{\Longrightarrow}  &  &
\X^{\sf op} \ar@{=}[rr] \ar@{<-}[dl]^{\dagger^\op} & \ar@{}[d]|{\Downarrow ~ \iota^{-1}} &
\X^{\sf op} \ar[dr]^{\dagger} \\
& \X \ar@{=}[rrrr]&  & & & \X }
\]
where the isomorphism $\omega: (\_)^{\dagger*} \to (\_)^{*\dagger}$ is from the cyclor preserving condition for Frobenius linear functors.

\[
\omega := \begin{tikzpicture}
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\node [style=none] (4) at (-2, 4) {};
\node [style=circle, scale=2.5] (5) at (-2, 2.25) {};
\node [style=none] (6) at (-2, 1) {};
\node [style=none] (7) at (-2, 2.25) {$\psi^\dagger$};
\node [style=none] (8) at (0.5, 4.5) {$X^{\dagger*}$};
\node [style=none] (9) at (-2.5, 1.5) {$X^{*\dagger}$};
\node [style=none] (10) at (-0.5, 2.25) {$\epsilon*$};
\node [style=none] (11) at (-1.5, 4.75) {$(*\epsilon)^\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
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\draw (4.center) to (5);
\draw (5) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~~~~~~ \omega^{-1} := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 1) {};
\node [style=none] (1) at (-2, 3) {};
\node [style=none] (2) at (-1, 3) {};
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\node [style=none] (4) at (0, 2) {};
\node [style=circle, scale=2.5] (5) at (0, 3.75) {};
\node [style=none] (6) at (0, 5) {};
\node [style=none] (7) at (0, 3.75) {$\psi^\dagger$};
\node [style=none] (8) at (-2.5, 1.5) {$X^{\dagger*}$};
\node [style=none] (9) at (0.5, 4.5) {$X^{*\dagger}$};
\node [style=none] (10) at (-1.5, 3.75) {$\eta*$};
\node [style=none] (11) at (-0.5, 1.25) {$(*\eta^{-1})^\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
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\draw (4.center) to (5);
\draw (5) to (6.center);
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\end{tikzpicture}
\]

It remains to show that the unit and the counit maps are inverses of each other in $\X$:

\[ (a) ~~~~~ %dbig2
\begin{tikzpicture}
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\node [style=none] (7) at (0, 3) {$\psi^{-1}$};
\node [style=none] (8) at (-1.5, 2.25) {$\epsilon*$};
\node [style=none] (9) at (-0.5, 4.75) {$*\eta$};
\node [style=none] (10) at (0, 2) {};
\node [style=none] (11) at (-1.75, 1.75) {$\eta*$};
\node [style=none] (12) at (-1, -1) {};
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\node [style=none] (16) at (-0.5, -1.75) {$\epsilon*$};
\node [style=none] (17) at (0, -1) {};
\node [style=none] (18) at (-3.5, -1.25) {};
\node [style=none] (19) at (-3.5, -0.25) {};
\node [style=none] (20) at (-4.5, -0.25) {};
\node [style=circle, scale=2.5] (21) at (-4.5, -1.25) {};
\node [style=none] (22) at (-4.5, -2.25) {};
\node [style=none] (23) at (-4.5, -1.25) {$\psi^\dagger$};
\node [style=circle, scale=2.5] (24) at (-1, -0) {};
\node [style=none] (25) at (-1, -0) {$\iota^{-1}$};
\node [style=none] (26) at (-5.5, -2) {$X^{*\dagger*\dagger}$};
\node [style=none] (27) at (-4, 0.5) {$(*\epsilon)^\dagger$};
\node [style=none] (28) at (-3, -2) {$\epsilon*$};
\node [style=none] (29) at (-1.75, 0.5) {$(X^*)^{\dagger \dagger}$};
\node [style=none] (30) at (-1.5, -0.75) {$X^*$};
\node [style=none] (31) at (0.5, 0.5) {$X^{**}$};
\node [style=none] (32) at (-2.5, 4.5) {$X$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend right=90, looseness=2.00] (1.center) to (2.center);
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\draw (4.center) to (5);
\draw (5) to (6.center);
\draw (10.center) to (17.center);
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\draw [bend right=90, looseness=1.75] (15.center) to (13.center);
\draw (14.center) to (13.center);
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\draw (18.center) to (19.center);
\draw [bend right=90, looseness=1.50] (19.center) to (20.center);
\draw (20.center) to (21);
\draw (21) to (22.center);
\draw (15.center) to (24);
\draw (24) to (12.center);
\end{pgfonlayer}
\end{tikzpicture} = %dbig2
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 5) {};
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\node [style=none] (6) at (0, 2) {};
\node [style=none] (7) at (0, 3) {$\psi^{-1}$};
\node [style=none] (8) at (-1.5, 2.25) {$\epsilon*$};
\node [style=none] (9) at (-0.5, 4.75) {$*\eta$};
\node [style=none] (10) at (0, 2) {};
\node [style=none] (11) at (-1, -1) {};
\node [style=none] (12) at (-2.5, 0.75) {};
\node [style=none] (13) at (-1, 0.75) {};
\node [style=none] (14) at (-0.5, -1.75) {$\epsilon*$};
\node [style=none] (15) at (0, -1) {};
\node [style=circle, scale=2.5] (16) at (-2.5, -1) {};
\node [style=none] (17) at (-2.5, -2) {};
\node [style=none] (18) at (-2.5, -1) {$\psi^\dagger$};
\node [style=circle, scale=2.5] (19) at (-1, -0) {};
\node [style=none] (20) at (-1, -0) {$\iota^{-1}$};
\node [style=none] (21) at (-3.5, -1.75) {$X^{*\dagger*\dagger}$};
\node [style=none] (22) at (-2, 1.75) {$(*\epsilon)^\dagger$};
\node [style=none] (23) at (-1.75, 0.5) {$(X^*)^{\dagger \dagger}$};
\node [style=none] (24) at (-1.5, -0.75) {$X^*$};
\node [style=none] (25) at (0.5, 0.5) {$X^{**}$};
\node [style=none] (26) at (-2.5, 4.5) {$X$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend right=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend left=90, looseness=1.75] (3.center) to (4.center);
\draw (4.center) to (5);
\draw (5) to (6.center);
\draw (10.center) to (15.center);
\draw [bend left=90, looseness=2.00] (15.center) to (11.center);
\draw [bend right=90, looseness=1.75] (13.center) to (12.center);
\draw (16) to (17.center);
\draw (13.center) to (19);
\draw (19) to (11.center);
\draw (12.center) to (16);
\end{pgfonlayer}
\end{tikzpicture} = %dbig3
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (2, 4.75) {};
\node [style=none] (1) at (2, 2.75) {};
\node [style=none] (2) at (1, 2.75) {};
\node [style=none] (3) at (1, 3.75) {};
\node [style=none] (4) at (0, 3.75) {};
\node [style=none] (5) at (0, 2) {};
\node [style=none] (6) at (1.5, 2) {$*\epsilon$};
\node [style=none] (7) at (0.5, 4.5) {$\eta*$};
\node [style=none] (8) at (0, 2) {};
\node [style=none] (9) at (-1, -1) {};
\node [style=none] (10) at (-2.5, 0.75) {};
\node [style=none] (11) at (-1, 0.75) {};
\node [style=none] (12) at (-0.5, -1.75) {$\epsilon*$};
\node [style=none] (13) at (0, -1) {};
\node [style=circle, scale=2.5] (14) at (-2.5, -1) {};
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\node [style=circle, scale=2.5] (17) at (-1, -0) {};
\node [style=none] (18) at (-1, -0) {$\iota^{-1}$};
\node [style=none] (19) at (-3.5, -1.75) {$X^{*\dagger*\dagger}$};
\node [style=none] (20) at (-2, 1.75) {$(*\epsilon)^\dagger$};
\node [style=none] (21) at (-1.75, 0.5) {$(X^*)^{\dagger \dagger}$};
\node [style=none] (22) at (-1.5, -0.75) {$X^*$};
\node [style=none] (23) at (0.5, 0.5) {$X^{**}$};
\node [style=none] (24) at (2.5, 4.25) {$X$};
\node [style=circle, scale=2.5] (25) at (1, 3.25) {};
\node [style=none] (26) at (1, 3.25) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
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\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
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\draw (8.center) to (13.center);
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\draw (14) to (15.center);
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\end{pgfonlayer}
\end{tikzpicture} = %dbig4
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 4.5) {};
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\node [style=none] (15) at (-1.75, 3) {$X^{*^{\dagger \dagger}}$};
\node [style=none] (16) at (-1.5, 1.75) {$X^*$};
\node [style=none] (17) at (0.5, 4) {$X$};
\node [style=circle, scale=2.5] (18) at (-1, 0.75) {};
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\draw (18) to (2.center);
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\end{tikzpicture} = %dbig5
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
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\end{tikzpicture} =
\]

\[
= %dbig6
\begin{tikzpicture}
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\node [style=none] (6) at (-2.5, -2) {};
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\node [style=none] (12) at (-1.75, 3) {$(X^*)^{\dagger \dagger}$};
\node [style=none] (13) at (-1.5, -0) {$(~^*X)^{\dagger \dagger}$};
\node [style=none] (14) at (0.75, 4) {$X$};
\node [style=circle, scale=2.5] (15) at (0.5, 1.5) {};
\node [style=none] (16) at (0.5, 1.5) {$\iota^{-1}$};
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\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
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\draw (0.center) to (15);
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\end{pgfonlayer}
\end{tikzpicture} \stackrel{*}{=} \begin{tikzpicture}
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\node [style=none] (13) at (-1.75, 1.75) {$(~^*X)^{\dagger}$};
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\node [style=none] (24) at (1.5, -1.25) {$\dagger$};
\node [style=none] (25) at (0.5, 4) {};
\node [style=none] (26) at (-2.5, -1.5) {};
\node [style=none] (27) at (-3.25, 0.25) {$~^*(X^{*\dagger})$};
\node [style=none] (28) at (-3, 2.5) {$X^{*\dagger*}$};
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\draw (3.center) to (5);
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\draw (8) to (2.center);
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\draw (15) to (25.center);
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\end{tikzpicture} \stackrel{\ref{Lemma:  cyclic dagger}}{=} %big8
\begin{tikzpicture}
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\node [style=none] (4) at (-1, 0.25) {};
\node [style=none] (5) at (-1, -3.25) {};
\node [style=none] (6) at (-1.75, -2.5) {$X^{*\dagger*\dagger}$};
\node [style=none] (7) at (-0.5, 5.75) {$X$};
\node [style=circle, scale=2.5] (8) at (-0.75, 5) {};
\node [style=none] (9) at (-0.75, 5) {$\iota$};
\node [style=none] (10) at (-1.5, 3.25) {$*\epsilon^\dagger$};
\node [style=none] (11) at (-3.75, 4) {};
\node [style=none] (12) at (1.75, 4) {};
\node [style=none] (13) at (1.75, -1.5) {};
\node [style=none] (14) at (-3.75, -1.5) {};
\node [style=none] (15) at (0.5, 4) {};
\node [style=none] (16) at (-2.5, -1.5) {};
\node [style=none] (17) at (1.5, -1.25) {$\dagger$};
\node [style=none] (18) at (-0.75, 4) {};
\node [style=none] (19) at (-1, -1.5) {};
\node [style=none] (20) at (-0.25, 4.5) {$X^{\dagger}$};
\node [style=none] (21) at (-3, -1) {$X^\dagger$};
\node [style=none] (22) at (1, 3.5) {$X^{*\dagger*}$};
\node [style=none] (23) at (-1.5, 1) {$(X^*)^\dagger$};
\node [style=none] (24) at (-0.25, -0.75) {$\epsilon*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw [bend right=90, looseness=1.75] (4.center) to (3.center);
\draw (0.center) to (8);
\draw (11.center) to (12.center);
\draw (12.center) to (13.center);
\draw (13.center) to (14.center);
\draw (14.center) to (11.center);
\draw (8) to (18.center);
\draw (19.center) to (5.center);
\draw (15.center) to (3.center);
\draw (2.center) to (4.center);
\draw (1.center) to (16.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

\[
(b) ~~~~~~ %bbig1
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 3.5) {};
\node [style=none] (1) at (-3, -2.5) {};
\node [style=none] (2) at (4.5, 3.5) {};
\node [style=none] (3) at (4.5, -2.5) {};
\node [style=none] (4) at (3, 3.5) {};
\node [style=none] (5) at (3, -1) {};
\node [style=none] (6) at (-0.25, -1) {};
\node [style=none] (7) at (-0.25, 1) {};
\node [style=none] (8) at (0.75, 1) {};
\node [style=none] (9) at (0.75, -0.25) {};
\node [style=none] (10) at (2, -0.25) {};
\node [style=circle, scale=2.5] (11) at (2, 0.75) {};
\node [style=none] (12) at (2, 1.75) {};
\node [style=none] (13) at (-2, 1.75) {};
\node [style=none] (14) at (-2, -2.5) {};
\node [style=none] (15) at (0, 3.25) {$\eta*$};
\node [style=none] (16) at (0.25, 1.75) {$\eta*$};
\node [style=none] (17) at (1.5, -1.25) {$(*\eta)^\dagger$};
\node [style=none] (18) at (1.5, -2.25) {$\epsilon*$};
\node [style=none] (19) at (3.5, 3) {$X^{\dagger**}$};
\node [style=none] (20) at (2.5, 1.5) {$X^{*\dagger}$};
\node [style=none] (21) at (2.5, -0) {$(^*X)^\dagger$};
\node [style=none] (22) at (0.25, 0.5) {$X^*$};
\node [style=none] (23) at (-0.75, -0.5) {$X^{\dagger*}$};
\node [style=none] (24) at (-2.5, -2) {$X^{*\dagger*}$};
\node [style=none] (25) at (1, 3.5) {};
\node [style=none] (26) at (1, 5.5) {};
\node [style=none] (27) at (1.5, 5) {$X^{*\dagger*\dagger}$};
\node [style=none] (28) at (2, 0.75) {$\psi^{-1^\dagger}$};
\node [style=none] (29) at (-2, -4) {};
\node [style=none] (30) at (-2, -6) {};
\node [style=none] (31) at (-0.75, -6) {};
\node [style=none] (32) at (-0.75, -5.5) {};
\node [style=none] (33) at (1, -5.5) {};
\node [style=none] (34) at (1, -8.5) {};
\node [style=circle, scale=2.5] (35) at (1, -7) {};
\node [style=none] (99) at (1, -7) {$\psi^{-1}$};
\node [style=none] (36) at (0, -4.5) {$*\eta$};
\node [style=none] (37) at (-1.5, -6.75) {$\epsilon*$};
\node [style=none] (38) at (1.5, -6.25) {$~^*(X^{\dagger*})$};
\node [style=none] (39) at (1.5, -8) {$X^{\dagger * *}$};
\node [style=none] (40) at (-0.25, -5.75) {$X^{\dagger*}$};
\node [style=none] (41) at (-2.5, -4.5) {$X^\dagger$};
\node [style=none] (42) at (-3, -4) {};
\node [style=none] (43) at (2.5, -4) {};
\node [style=none] (44) at (2.5, -8.5) {};
\node [style=none] (45) at (-3, -8.5) {};
\node [style=none] (46) at (0, -2.5) {};
\node [style=none] (47) at (0, -4) {};
\node [style=none] (48) at (0.5, -3.25) {$X^{\dagger**\dagger}$};
\node [style=circle, scale=2.5] (49) at (0, -9.5) {};
\node [style=none] (50) at (0, -10.5) {};
\node [style=none] (51) at (0, -8.5) {};
\node [style=none] (52) at (0, -9.5) {$\iota^{-1}$};
\node [style=none] (53) at (-0.5, -9) {$X^{\dagger \dagger}$};
\node [style=none] (54) at (-0.5, -10.25) {$X$};
\node [style=none] (55) at (4, -2) {$\dagger$};
\node [style=none] (56) at (2, -8.25) {$\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.00] (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw [bend left=90, looseness=1.50] (7.center) to (8.center);
\draw (8.center) to (9.center);
\draw [bend right=90, looseness=1.50] (9.center) to (10.center);
\draw (10.center) to (11);
\draw (11) to (12.center);
\draw [bend right=90, looseness=1.00] (12.center) to (13.center);
\draw (13.center) to (14.center);
\draw (26.center) to (25.center);
\draw (29.center) to (30.center);
\draw [bend right=90, looseness=1.50] (30.center) to (31.center);
\draw (31.center) to (32.center);
\draw [bend left=90, looseness=1.25] (32.center) to (33.center);
\draw (33.center) to (35);
\draw (35) to (34.center);
\draw (43.center) to (44.center);
\draw (44.center) to (45.center);
\draw (45.center) to (42.center);
\draw (42.center) to (43.center);
\draw (46.center) to (47.center);
\draw (51.center) to (49);
\draw (49) to (50.center);
\end{pgfonlayer}
\end{tikzpicture} = %bbig2
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2.5) {};
\node [style=none] (1) at (-3, -8.5) {};
\node [style=none] (2) at (4.5, 2.5) {};
\node [style=none] (3) at (4.5, -8.5) {};
\node [style=none] (4) at (3, -2) {};
\node [style=none] (5) at (3, -7) {};
\node [style=none] (6) at (-0.25, -7) {};
\node [style=none] (7) at (-0.25, -5) {};
\node [style=none] (8) at (0.75, -5) {};
\node [style=none] (9) at (0.75, -6.25) {};
\node [style=none] (10) at (2, -6.25) {};
\node [style=circle, scale=2.5] (11) at (2, -5.25) {};
\node [style=none] (12) at (2, -4.25) {};
\node [style=none] (13) at (-2, -4.25) {};
\node [style=none] (14) at (-2, -8.5) {};
\node [style=none] (15) at (0, -2.75) {$\eta*$};
\node [style=none] (16) at (0.25, -4.25) {$\eta*$};
\node [style=none] (17) at (1.5, -7.25) {$(*\eta)^\dagger$};
\node [style=none] (18) at (1.5, -8.25) {$\epsilon*$};
\node [style=none] (19) at (2.5, -4.5) {$X^{*\dagger}$};
\node [style=none] (20) at (2.5, -6) {$(^*X)^\dagger$};
\node [style=none] (21) at (0.25, -5.5) {$X^*$};
\node [style=none] (22) at (-0.75, -6.5) {$X^{\dagger*}$};
\node [style=none] (23) at (-2.5, -8) {$X^{*\dagger*}$};
\node [style=none] (24) at (0.9999999, 2.5) {};
\node [style=none] (25) at (0.9999999, 4.5) {};
\node [style=none] (26) at (1.5, 4) {$X^{*\dagger*\dagger}$};
\node [style=none] (27) at (2, -5.25) {$\psi^{-1^\dagger}$};
\node [style=none] (28) at (0.5, 1.75) {$X^\dagger$};
\node [style=none] (29) at (0, -8.5) {};
\node [style=circle, scale=2.5] (30) at (0, -11) {};
\node [style=none] (31) at (0, -12.25) {};
\node [style=none] (32) at (0, -11) {$\iota^{-1}$};
\node [style=none] (33) at (-0.9999999, -9.75) {$X^{\dagger \dagger}$};
\node [style=none] (34) at (-0.4999999, -12) {$X$};
\node [style=none] (35) at (4, -8) {$\dagger$};
\node [style=none] (36) at (1.25, 1) {};
\node [style=none] (37) at (1.25, 0.4999999) {};
\node [style=none] (38) at (0, 2.5) {};
\node [style=circle, scale=2.5] (39) at (3, -0.7499999) {};
\node [style=none] (39) at (3, -0.7499999) {$\psi^{-1}$};
\node [style=none] (40) at (0, 0.4999999) {};
\node [style=none] (41) at (1.75, 0.7500001) {$X^{\dagger*}$};
\node [style=none] (42) at (2, 2) {$*\eta$};
\node [style=none] (43) at (3, 1) {};
\node [style=none] (44) at (3.5, 0.25) {$~^*(X^{\dagger*})$};
\node [style=none] (45) at (3, -2) {};
\node [style=none] (46) at (0.4999999, -0.25) {$\epsilon*$};
\node [style=none] (47) at (3.5, -2.75) {$X^{\dagger * *}$};
\node [style=none] (48) at (2, 2.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.00] (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw [bend left=90, looseness=1.50] (7.center) to (8.center);
\draw (8.center) to (9.center);
\draw [bend right=90, looseness=1.50] (9.center) to (10.center);
\draw (10.center) to (11);
\draw (11) to (12.center);
\draw [bend right=90, looseness=1.00] (12.center) to (13.center);
\draw (13.center) to (14.center);
\draw (25.center) to (24.center);
\draw (30) to (31.center);
\draw (38.center) to (40.center);
\draw [bend right=90, looseness=1.50] (40.center) to (37.center);
\draw (37.center) to (36.center);
\draw [bend left=90, looseness=1.25] (36.center) to (43.center);
\draw (43.center) to (39);
\draw (39) to (45.center);
\draw (29.center) to (30);
\end{pgfonlayer}
\end{tikzpicture} = %bbig3
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2.5) {};
\node [style=none] (1) at (-3, -8.5) {};
\node [style=none] (2) at (6.75, 2.5) {};
\node [style=none] (3) at (6.75, -8.5) {};
\node [style=none] (4) at (-0.25, -7) {};
\node [style=none] (5) at (-0.25, -5) {};
\node [style=none] (6) at (0.75, -5) {};
\node [style=none] (7) at (0.75, -6.25) {};
\node [style=none] (8) at (2, -6.25) {};
\node [style=circle, scale=2.5] (9) at (2, -5.25) {};
\node [style=none] (10) at (2, -4.25) {};
\node [style=none] (11) at (-2, -4.25) {};
\node [style=none] (12) at (-2, -8.5) {};
\node [style=none] (13) at (0, -2.75) {$\eta*$};
\node [style=none] (14) at (0.25, -4.25) {$\eta*$};
\node [style=none] (15) at (1.5, -7.25) {$(*\eta)^\dagger$};
\node [style=none] (16) at (1.5, -8.25) {$\epsilon*$};
\node [style=none] (17) at (0.25, -5.5) {$X^*$};
\node [style=none] (18) at (-0.75, -6.5) {$X^{\dagger*}$};
\node [style=none] (19) at (-2.5, -8) {$X^{*\dagger*}$};
\node [style=none] (20) at (0.9999999, 2.5) {};
\node [style=none] (21) at (0.9999999, 4.5) {};
\node [style=none] (22) at (1.5, 4) {$X^{*\dagger*\dagger}$};
\node [style=none] (23) at (2, -5.25) {$\psi^{-1^\dagger}$};
\node [style=none] (24) at (0, -8.5) {};
\node [style=circle, scale=2.5] (25) at (0, -11) {};
\node [style=none] (26) at (0, -12.25) {};
\node [style=none] (27) at (0, -11) {$\iota^{-1}$};
\node [style=none] (28) at (-0.9999999, -9.75) {$X^{\dagger \dagger}$};
\node [style=none] (29) at (-0.4999999, -12) {$X$};
\node [style=none] (30) at (6.25, -8) {$\dagger$};
\node [style=none] (31) at (2, 2.5) {};
\node [style=none] (32) at (4.5, 1.25) {};
\node [style=circle, scale=2.5] (33) at (4.5, -0) {};
\node [style=none] (34) at (4.5, -0.9999999) {};
\node [style=none] (35) at (5.75, -0.9999999) {};
\node [style=none] (36) at (5.75, 2.5) {};
\node [style=none] (37) at (4.5, -0) {$\psi$};
\node [style=none] (38) at (5, -2) {$*\epsilon$};
\node [style=none] (39) at (5.25, 2) {$X^\dagger$};
\node [style=none] (40) at (4, 0.7499999) {$(X^{\dagger*}$};
\node [style=none] (41) at (4, -0.9999999) {$~^*(X^\dagger)$};
\node [style=none] (42) at (3, 1.25) {};
\node [style=none] (43) at (2.5, -6) {$(^*X)^\dagger$};
\node [style=none] (44) at (3, -2) {};
\node [style=none] (45) at (2.5, -4.5) {$X^{*\dagger}$};
\node [style=none] (46) at (3.5, -2.75) {$X^{\dagger * *}$};
\node [style=none] (47) at (3.75, 2) {$\eta*$};
\node [style=none] (48) at (3, -7) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.50] (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw [bend right=90, looseness=1.50] (7.center) to (8.center);
\draw (8.center) to (9);
\draw (9) to (10.center);
\draw [bend right=90, looseness=1.00] (10.center) to (11.center);
\draw (11.center) to (12.center);
\draw (21.center) to (20.center);
\draw (25) to (26.center);
\draw (24.center) to (25);
\draw (32.center) to (33);
\draw (33) to (34.center);
\draw [bend right=90, looseness=2.00] (34.center) to (35.center);
\draw (35.center) to (36.center);
\draw [bend left=90, looseness=1.00] (48.center) to (4.center);
\draw (42.center) to (48.center);
\draw [bend left=90, looseness=1.25] (42.center) to (32.center);
\end{pgfonlayer}
\end{tikzpicture} =  \]

\[ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2.5) {};
\node [style=none] (1) at (-3, -8.5) {};
\node [style=none] (2) at (4.25, 2.5) {};
\node [style=none] (3) at (4.25, -8.5) {};
\node [style=none] (4) at (-0.25, -3.25) {};
\node [style=none] (5) at (-0.25, -0.7499999) {};
\node [style=none] (6) at (0.7499999, -0.7499999) {};
\node [style=none] (7) at (0.7499999, -2) {};
\node [style=none] (8) at (2, -2) {};
\node [style=circle, scale=2.5] (9) at (2, -0.9999999) {};
\node [style=none] (10) at (2, -0) {};
\node [style=none] (11) at (-2, -0) {};
\node [style=none] (12) at (-2, -8.5) {};
\node [style=none] (13) at (0, 1.5) {$\eta*$};
\node [style=none] (14) at (0.25, -0) {$\eta*$};
\node [style=none] (15) at (1.5, -2.75) {$(*\eta)^\dagger$};
\node [style=none] (16) at (0.25, -1.25) {$X^*$};
\node [style=none] (17) at (-0.7499999, -2.25) {$X^{\dagger*}$};
\node [style=none] (18) at (-2.5, -8) {$X^{*\dagger*}$};
\node [style=none] (19) at (0.9999999, 2.5) {};
\node [style=none] (20) at (0.9999999, 4.5) {};
\node [style=none] (21) at (1.5, 4) {$X^{*\dagger*\dagger}$};
\node [style=none] (22) at (2, -0.9999999) {$\psi^{-1^\dagger}$};
\node [style=none] (23) at (0, -8.5) {};
\node [style=circle, scale=2.5] (24) at (0, -11) {};
\node [style=none] (25) at (0, -12.25) {};
\node [style=none] (26) at (0, -11) {$\iota^{01}$};
\node [style=none] (27) at (-0.9999999, -9.75) {$X^{\dagger \dagger}$};
\node [style=none] (28) at (-0.4999999, -12) {$X$};
\node [style=none] (29) at (3.75, -8) {$\dagger$};
\node [style=none] (30) at (2, 2.5) {};
\node [style=circle, scale=2.5] (31) at (2, -5.75) {};
\node [style=none] (32) at (2, -6.25) {};
\node [style=none] (33) at (3.25, -6.25) {};
\node [style=none] (34) at (3.25, 2.5) {};
\node [style=none] (35) at (2, -5.75) {$\psi$};
\node [style=none] (36) at (2.5, -7.25) {$*\epsilon$};
\node [style=none] (37) at (2.75, 2) {$X^\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.50] (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw [bend right=90, looseness=1.50] (7.center) to (8.center);
\draw (8.center) to (9);
\draw (9) to (10.center);
\draw [bend right=90, looseness=1.00] (10.center) to (11.center);
\draw (11.center) to (12.center);
\draw (20.center) to (19.center);
\draw (24) to (25.center);
\draw (23.center) to (24);
\draw (31) to (32.center);
\draw [bend right=90, looseness=2.00] (32.center) to (33.center);
\draw (33.center) to (34.center);
\draw [in=90, out=-90, looseness=0.75] (4.center) to (31);
\end{pgfonlayer}
\end{tikzpicture} \stackrel{*}{=}%bbig5
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2.5) {};
\node [style=none] (1) at (-3, -8.5) {};
\node [style=none] (2) at (4.5, 2.5) {};
\node [style=none] (3) at (4.5, -8.5) {};
\node [style=none] (4) at (2, -1.5) {};
\node [style=none] (5) at (2, -0.7499999) {};
\node [style=none] (6) at (1, -0.7499999) {};
\node [style=none] (7) at (1, -2) {};
\node [style=none] (8) at (-0.25, -2) {};
\node [style=none] (9) at (-0.25, 0.7500001) {};
\node [style=none] (10) at (-2, 0.7500001) {};
\node [style=none] (11) at (-2, -8.5) {};
\node [style=none] (12) at (-0.9999999, 1.5) {$\eta*$};
\node [style=none] (13) at (-2.5, -8) {$X^{*\dagger*}$};
\node [style=none] (14) at (0.9999999, 2.5) {};
\node [style=none] (15) at (0.9999999, 4.5) {};
\node [style=none] (16) at (1.5, 4) {$X^{*\dagger*\dagger}$};
\node [style=none] (17) at (0, -8.5) {};
\node [style=circle, scale=2.5] (18) at (0, -11) {};
\node [style=none] (19) at (0, -12.25) {};
\node [style=none] (20) at (0, -11) {$\iota^{-1}$};
\node [style=none] (21) at (-0.9999999, -9.75) {$X^{\dagger \dagger}$};
\node [style=none] (22) at (-0.4999999, -12) {$X$};
\node [style=none] (23) at (4, -8) {$\dagger$};
\node [style=none] (24) at (2, 2.5) {};
\node [style=circle, scale=2.5] (25) at (2, -5.75) {};
\node [style=none] (26) at (2, -6.25) {};
\node [style=none] (27) at (3.5, -6.25) {};
\node [style=none] (28) at (3.5, 2.5) {};
\node [style=none] (29) at (2, -5.75) {$\psi$};
\node [style=none] (30) at (2.75, -7) {$*\epsilon$};
\node [style=none] (31) at (3, 2) {$X^\dagger$};
\node [style=circle, scale=2.5] (32) at (2, -3.25) {};
\node [style=none] (33) at (2, -5) {};
\node [style=none] (34) at (2, -3.25) {$\psi^{-1}$};
\node [style=none] (35) at (0.25, -2.75) {$(\eta*)^\dagger$};
\node [style=none] (36) at (1.5, -0) {$\eta*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend right=90, looseness=1.50] (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw [bend left=90, looseness=1.50] (7.center) to (8.center);
\draw [bend right=90, looseness=1.00] (9.center) to (10.center);
\draw (10.center) to (11.center);
\draw (15.center) to (14.center);
\draw (18) to (19.center);
\draw (17.center) to (18);
\draw (25) to (26.center);
\draw [bend right=90, looseness=0.75] (26.center) to (27.center);
\draw (27.center) to (28.center);
\draw (4.center) to (32);
\draw (32) to (33.center);
\draw (9.center) to (8.center);
\draw (33.center) to (25);
\end{pgfonlayer}
\end{tikzpicture} = %bbig6
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, 2.5) {};
\node [style=none] (1) at (-2.5, -8.5) {};
\node [style=none] (2) at (4.75, 2.5) {};
\node [style=none] (3) at (4.75, -8.5) {};
\node [style=none] (4) at (2.5, -3.75) {};
\node [style=none] (5) at (2.5, -5) {};
\node [style=none] (6) at (0.7499999, -5) {};
\node [style=none] (7) at (0.7499999, 0.4999999) {};
\node [style=none] (8) at (-1.5, 0.7499999) {};
\node [style=none] (9) at (-1.5, -8.5) {};
\node [style=none] (10) at (-0.4999999, 1.5) {$\eta*$};
\node [style=none] (11) at (-2, -8) {$X^{*\dagger*}$};
\node [style=none] (12) at (0.9999999, 2.5) {};
\node [style=none] (13) at (0.9999999, 4.5) {};
\node [style=none] (14) at (1.5, 4) {$X^{*\dagger*\dagger}$};
\node [style=none] (15) at (0, -8.5) {};
\node [style=circle, scale=2.5] (16) at (0, -11) {};
\node [style=none] (17) at (0, -12.25) {};
\node [style=none] (18) at (0, -11) {$\iota^{-1}$};
\node [style=none] (19) at (-0.9999999, -9.75) {$X^{\dagger \dagger}$};
\node [style=none] (20) at (-0.4999999, -12) {$X$};
\node [style=none] (21) at (4.25, -8) {$\dagger$};
\node [style=none] (22) at (2.5, -3.75) {};
\node [style=none] (23) at (2.5, 2.5) {};
\node [style=none] (24) at (2, 2) {$X^\dagger$};
\node [style=none] (25) at (1.75, -6.25) {$(\eta*)^\dagger$};
\node [style=none] (26) at (1.25, -2.5) {$X^{*\dagger}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.75] (5.center) to (6.center);
\draw [bend right=90, looseness=1.00] (7.center) to (8.center);
\draw (8.center) to (9.center);
\draw (13.center) to (12.center);
\draw (16) to (17.center);
\draw (15.center) to (16);
\draw (22.center) to (23.center);
\draw (7.center) to (6.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

$(*)$ holds because $\dagger$ preserves the cyclor. Thus, $(a)$ and $(b)$ are inverses of each other.

§ UNITARY STRUCTURE AND MIXED UNITARY CATEGORIES

The objective of this section is to introduce mixed unitary categories (MUCs) and their morphisms.   A mixed unitary category consist of
a unitary category, $\U$, with a $\dagger$-isomix Frobenius functor $V$ into the core of a “large” isomix category, so that $V: \U \to {\sf Core}(\X) \to \X$.
The unitary category is to be regarded as providing the analogue of scalars for the larger category much as a field provides scalars
for an algebra over that field.

The section starts by describing the general notion of unitary structure in a $\dagger$-isomix category. This allows the definition of a unitary category as
a compact $\dagger$-isomix category in which all objects have unitary structure.   We then show how to extract a unitary category from any $\dagger$-isomix
category using pre-unitary objects (introduced below).  This is a useful construction, however, it should be born in mind that it can deliver a trivial unitary
category – trivial in the sense that all objects are isomorphic to the units.  This means that, in any application of the construction, it is important to identify
non-trivial pre-unitary objects to ensure that one is getting something worthwhile out.

Next we show, using the isomix functors ${\sf Mx}_\uparrow$ (or ${\sf Mx}_\downarrow$) that unitary categories are $\dagger$-linearly equivalent to
$\dagger$-monoidal categories and, furthermore, that closed unitary categories are equivalent to $\dagger$-compact closed categories.  This provides an
explicit connection to the standard notions from categorical quantum mechanics.   One contribution of this more general perspective is that through the
constructions in this section one can obtain examples not only of mix unitary categories but also of $\dagger$-monoidal and $\dagger$-compact closed
categories which might otherwise have been difficult to realize.

The final subsection introduces the notion of a mixed unitary category which forms the basis for our approach to infinite dimensional categorical quantum

§.§ Unitary structure

We are now ready to introduce the key notion of “unitary” which is natural for the setting of $\dagger$-isomix categories.  It will transpire that one can only have
unitary isomorphisms  between certain object, which we call “unitary objects”, and that these must be in the core of the isomix category:

A $\dagger$-isomix category, $\X$, has unitary structure in case there is an essentially small class of objects, ${\cal U}$, called the unitary objects of $\X$ such that:

* Every unitary object is in the core

* Each unitary object $A$ comes equipped with an isomorphism, called the unitary structure map of $A$, $\varphi_A: A \to A^\dag$

* Unitary objects are closed to $(\_)^\dag$ so that $\varphi_{A^\dag} = ((\varphi_A)^{-1})^\dag$

* The following diagram commutes:

\[   \xymatrix{  A   \ar[d]_{\varphi_A} \ar[drrr]^{\iota}  & \\ A^\dag \ar[rrr]_{\varphi_{A^\dag}}  & & & (A^\dag)^\dag  } \]

* The monoidal units $\top$ and $\bot$  are unitary objects and the following diagrams commute:

\[   \begin{matrix} \xymatrix{  \bot \ar[d]_{\m} \ar[dr]^{\varphi_\bot} \ar[r]^{\lambda_\bot} & \top ^\dagger \ar[d]^{(\m)^{\dagger}} \\  \top \ar[r]_{\lambda_\top} & \bot^\dag } \end{matrix}~~~~~
\begin{matrix} \xymatrix{  \top \ar[d]_{\m^{-1}} \ar[dr]^{\varphi_\top} \ar[r]^{\lambda_\top} & \bot ^\dagger \ar[d]^{(\m^{-1})^{\dagger}} \\  \bot \ar[r]_{\lambda_\bot} & \top^\dag } \end{matrix}
\]

* If $A$ and $B$ are unitary objects then $A \ox B$ and $A \oa B$ are unitary objects and the following diagrams commute:

\[  \begin{matrix} \xymatrix{A\ox B   \ar[r]^{\varphi_A\ox \varphi_B}  \ar[d]_{\mx} & A^\dag \ox B^\dag  \ar[d]^{\lambda_\ox}\\
A\oa B   \ar[r]_{\varphi_{A\oa B}} & (A\oa B)^\dag         } \end{matrix}~~~~~
\begin{matrix} \xymatrix{A\ox B   \ar[r]^{\varphi_{A\ox B}}  \ar[d]_{\mx}  & (A \ox B)^\dag  \\
A\oa B  \ar[r]_{\varphi_{A}\oa \varphi_B} & A^\dag\oa B^\dag  \ar@{->}[u]_{\lambda_\oa} } \end{matrix}
\]

A $\dagger$-isomix category with unitary structure in which all objects are unitary is a unitary category.

Unitary structure is structure, thus, a given category can have many different unitary structures.   The requirements, however, do mean that there is always a smallest unitary
structure which generated from the requirement that the tensor units be present: this we refer to this as the “trivial” unitary structure.  Our next observations show how this is generated.

If $A = \bot$, then by [U.5](a) $\varphi_\bot = \m \lambda_\top$, and by  [U.3], $\varphi_{\bot^\dagger} = ((\m \lambda_\top)^{-1})^\dagger$, and this means [U.4] hold
automatically as:

\begin{eqnarray*}
\m \lambda_\top ((\m \lambda_\top)^{-1})^\dagger
&=& \m \lambda_\top (\lambda_\top^{-1} \m^{-1})^\dagger \\
&=& \m \lambda_\top (\m^{-1})^\dag (\lambda_\top^{-1})^\dag \\
&\stackrel{\text{ \tiny{ \bf{[$\dagger$-mix]}}}}{=}& \m \m^{-1} \lambda_\bot (\lambda_\top^{-1})^\dag  \\ &\stackrel{\text{ \tiny{ \bf{[$\dagger$-ldc.6(a)]}}}}{=}& \iota
\end{eqnarray*}

Similarly, if $A = A \oa B$, then by [U.6](a) $\varphi_{A \oa B} = {\mx^{-1} }\varphi_A \ox \varphi_B \lambda_\ox$, and by  [U.3], $\varphi_{\bot^\dagger} = (({\mx^{-1} }\varphi+A \ox \varphi_B \lambda_\ox)^{-1})^\dagger$, and again [U.4] holds as:

When $A$ and $B$ are unitary objects in a $\dagger$-isomix category then

* ${\mx^{-1} }(\varphi_A \ox \varphi_B) \lambda_\ox (({\mx^{-1} }(\varphi_A \ox \varphi_B) \lambda_\ox)^{-1})^\dagger = \iota$;

* $\varphi_{A^{\dagger\dagger}} = (\varphi_A)^{\dagger \dagger}: A^{\dagger\dagger} \to A^{\dagger \dagger \dagger}$.

\begin{eqnarray*}
\lefteqn{ {\mx^{-1} }(\varphi_A \ox \varphi_B) \lambda_\ox (({\mx^{-1} }(\varphi_A \ox \varphi_B) \lambda_\ox)^{-1})^\dagger} \\
&=& {\mx^{-1} }(\varphi_A \ox \varphi_B) \lambda_\ox \mx^\dagger (\varphi_A^{-1} \ox  \varphi_B^{-1})^\dagger (\lambda_\ox^{-1})^\dagger \\
&\stackrel{\text{nat. } \mx}{=}& {\mx^{-1} }(\varphi_A \ox \varphi_B) \mx ~ \lambda_\oa (\varphi_A^{-1} \ox  \varphi_B^{-1})^\dagger (\lambda_\ox^{-1})^\dagger \\
&\stackrel{\text{nat. } \mx}{=}& (\varphi_A \oa \varphi_B) \lambda_\oa (\varphi_A^{-1} \ox  \varphi_B^{-1})^\dagger (\lambda_\ox^{-1})^\dagger \\
& \stackrel{\text{nat} ~\lambda_\oa}{=}& (\varphi_A)((\varphi_A)^{-1})^\dag (\varphi_B)((\varphi_B)^{-1})^\dag \lambda_\oa (\lambda_\ox^{-1})^\dagger \\
& \stackrel{\text{\tiny {\bf [U.3] \& [U.4]}}}{=}& (\iota \oa \iota) \lambda_\oa (\lambda_\ox^{-1})^\dagger \\
& \stackrel{\text{\tiny {\bf [$\dagger$-ldc.6](a)}}}{=}& \iota
\end{eqnarray*}

* We have:

\[ \varphi_{(A^\dagger)^{\dagger}} = ((\varphi_A^\dagger)^{-1})^{\dagger} = ((((\varphi_A)^{-1})^\dagger)^{-1})^\dagger = ((((\varphi_A)^{-1})^{-1})^\dagger)^\dagger = ((\varphi_A)^\dagger)^\dagger \]

These observations show that the definition of the unitary structure for objects can be inductively pushed down onto its tensor and par components and, thus, if such exists, onto atomic objects.
In particular, objects generated from the units will therefore have their unitary structure completely determined.  This shows that these objects with their unitary structure give the smallest unitary structure possible on the category (which we have called the trivial unitary structure).

Often we shall want the unitary objects to have linear adjoints (or duals) but we shall need the analogue of $\dagger$-duals from categorical quantum mechanics:

A unitary linear dual $(\eta, \epsilon): A \dashvv_{~u} B$ is a linear dual, $A \dashvv B$ with $A$ and $B$ being unitary objects satisfying in addition:

\[
\xymatrix{
\top \ar@{}[ddrr]|{(a)} \ar[rr]^{\eta_A} \ar[d]_{\lambda_\top}  & & A \oa B \ar[d]^{\varphi_A \oa \varphi_B} \\
\bot^\dagger \ar[d]_{\epsilon^\dag} & & A^\dagger \oa B^\dagger \ar[d]^{c_\oa} \\
(B \ox A)^\dag \ar[rr]_{\lambda_\oa^{-1}} & & B^\dagger \oa A^\dagger}
\xymatrix{
A \ox B \ar@{}[ddrr]|{(b)} \ar[rr]^{\varphi_A \ox \varphi_B} \ar[d]_{c_\ox} & & A^\dag \ox B^\dag \ar[d]^{\lambda_\ox} \\
B \ox A \ar[d]_{\epsilon_A} & & (A \oa B)^\dagger \ar[d]^{\eta_A^\dagger} \\
\bot \ar[rr]_{\lambda_\bot} & & \top^\dagger }
\]

A unitary category in which every unitary object has a unitary linear dual is a called a closed unitary category.  We shall see a more precise correspondence to categorical
quantum mechanical $\dagger$-duals in Proposition <ref>.

$\top  \dashvv_{~u} \bot$ and whenever $(\eta_1, \epsilon_1): V_1 \dashvv_{~u} U_1$ and $(\eta_2, \epsilon_2): V_2 \dashvv_{~u} U_2$ then $(V_1 \ox V_2) \dashvv_{~u} (U_1 \oa U_2)$ .

Define $(\eta', \epsilon'): (V_1 \ox V_2) \dashvv_{~u} (U_1 \oa U_2)$ where
$\eta' = \begin{tikzpicture} %opluseta
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-4, 3) {$\eta_1$};
\node [style=circle] (1) at (-2, 3) {$\eta_2$};
\node [style=ox] (2) at (-4, 1.75) {};
\node [style=oa] (3) at (-2, 1.75) {};
\node [style=none] (4) at (-4, 1) {};
\node [style=none] (5) at (-2, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=none, in=15, out=-165, looseness=1.00] (1) to (2);
\draw [style=none, bend left, looseness=1.25] (1) to (3);
\draw [style=none, in=180, out=-15, looseness=1.00] (0) to (3);
\draw [style=none, bend left=45, looseness=1.25] (2) to (0);
\draw [style=none] (2) to (4.center);
\draw [style=none] (3) to (5.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~
\epsilon' = \begin{tikzpicture} %oplusepsi
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-4, 1) {$\epsilon_1$};
\node [style=circle] (1) at (-2, 1) {$\epsilon_2$};
\node [style=oa] (2) at (-4, 2.25) {};
\node [style=ox] (3) at (-2, 2.25) {};
\node [style=none] (4) at (-4, 3) {};
\node [style=none] (5) at (-2, 3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=none, in=-15, out=165, looseness=1.00] (1) to (2);
\draw [style=none, bend right, looseness=1.25] (1) to (3);
\draw [style=none, in=180, out=15, looseness=1.00] (0) to (3);
\draw [style=none, bend right=45, looseness=1.25] (2) to (0);
\draw [style=none] (2) to (4.center);
\draw [style=none] (3) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
then this is easily checked to be a unitary linear adjoint.

This means a unitary category alway has a unitary nucleus which is a unitary category consisting of those objects with unitary duals.

Finally we can define what it means for an isomorphism to be unitary:

Suppose $A$ and $B$ are unitary objects in a $\dagger$-isomix category with unitary structure. An isomorphism $A \to^f B$ between unitary objects is said to be unitary if the following diagram commutes:

\[  \xymatrix{A   \ar[r]^{\varphi_A}    \ar[d]_{f} \ar[r]^{\varphi_A} & A^\dag \ar@{<-}[d]^{f^\dag} \\ B  \ar[r]_{\varphi_B} & B^\dag    }  \]

Observe that $\varphi$ is “twisted” natural for all unitary isomorphisms, thus, it is easily seen that unitary isomorphisms compose and identity maps are unitary.   Our next objective is to show that all the coherence isomorphisms between unitary objects are unitary maps when one has unitary structure.  In particular the linear distributors on unitary objects are isomorphisms as
everything is in the core and they are unitary isomorphisms.

First a warm up:

In a $\dagger$-isomix category with unitary structure:

* If $f$ is unitary, then so is $f^\dagger$;

* If $f$ and $g$ are unitary, then so are $f \ox g$ and $f \oa g$;

* Unitary isomorphisms are closed to composition.

* Recall that $\varphi_{A^\dag} = (\varphi_A^{-1})^\dag$  then $f^\dagger$ is unitary because

\[ \xymatrix{B^\dag \ar[d]_{(\varphi_B^{-1})^\dag = \varphi_{B^\dag}} \ar[rr]^{f^\dag} & & A^\dag \ar[d]^{(\varphi_A^{-1})^\dag = \varphi_{A^\dag}} \\
B^{\dag\dag}  & & A^{\dag\dag} \ar[ll]^{f^{\dag\dag}}} \]

is just the dagger functor applied to the unitary diagram of $f$.

* Suppose $f$ and $g$ are unitary morphisms, then:

\[
\xymatrix{
A \ox B \ar@{->}[rrr]^{\varphi_{A \ox B}} \ar[ddd]_{f \ox g} \ar[dr]_{\mx}  \ar@{}[dddr]|{\mbox{\tiny \bf (nat. $\mx$)}~~~}
&   \ar@{}[dr]|{\mbox{\tiny {\bf [U.6(b)]}}} &  &  (A \ox B)^\dagger \ar@{}[lddd]|{~~~~~\mbox{\tiny \bf (nat. $\lambda_\oa)$}} \\
& A \oa B \ar[r]^{\varphi_A \oa \varphi_B} \ar[d]_{f \oa g}
& A^\dagger \oa B^\dagger \ar[ur]_{\lambda_\oa}  &
\\ & A' \oa B' \ar[r]_{\varphi_{A'} \ox \varphi_{B'}} \ar@{}[dr]|{\mbox{\tiny { \bf [U.6(b)]}}}
& A'^\dagger \oa B'^\dagger \ar[dr]^{\lambda_\oa} \ar[u]_{f^\dagger \oa g^\dagger}
& \\ A' \ox B' \ar[rrr]_{\varphi_{A' \ox B'}} \ar[ur]_{\mx}
& & & (A' \ox B')^\dagger \ar[uuu]_{(f \ox g)^\dagger}
\]

The inner square commutes because $f$ and $g$ are unitary maps.
Similarly, using coherence [U.6(b)], one can show that if $f$ and $g$ are unitary, then $f \oa g$ is unitary.

* The proof is trivial.

Suppose $\X$ is a $\dagger$-isomix category with unitary structure and $A$, $B$, and $C$ are unitary objects then the following are unitary maps:

* $\lambda_\ox: A^\dagger \ox B^\dagger \rightarrow (A \oa B)^\dagger$

* $\lambda_\oa:  A^\dagger \oa B^\dagger \rightarrow (A \ox B)^\dagger$

* $\lambda_\top: \top \rightarrow \bot^\dagger$

* $\lambda_\bot: \bot \to \top^\dagger$

* $\varphi_A: A \rightarrow A^\dagger$

* $m: \top \rightarrow \bot$

* $\mx_{A,B}: A \ox B \rightarrow A \oa B$

* $\iota : A \rightarrow (A^{\dagger})^\dagger$

* $a_\ox: A \ox (B \ox C) \rightarrow (A \ox B) \ox C$

* $a_\oa: A \oa ( B \oa C) \rightarrow (A \oa B) \oa C$

* $c_\ox: A \ox B \rightarrow B \ox A$

* $c_\oa: A \oa B \rightarrow B \oa A$

* $\partial_L: A \ox (B \oa C) \rightarrow (A \ox B) \oa C$

* $\partial_R: (A \oa R) \ox C \rightarrow A \oa (B \ox C)$

* $\lambda_\ox: A^\dagger \ox B^\dagger \rightarrow (A \oa B)^\dagger$ is a unitary map because:

\[\xymatrixcolsep{4pc}\xymatrix{
A^\dag\ox B^\dag \ar[r]^{\phi_A^{-1}\ox\phi_B^{-1}} \ar[d]^{\lambda_\ox} \ar@{=}@/^3pc/[rr]     \ar@{}[dr]|{\mbox{\tiny {\bf [U.6(a)] }}}
&  A\ox B \ar[r]^{\phi_A\ox \phi_B}  \ar[d]^{\mx} 	                     \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
&  A^\dag \ox B^\dag \ar[r]^{\phi_{A^\dag \ox B^\dag}} \ar[d]^{\mx}									 \ar@{}[dr]|{\mbox{\tiny {\bf [U.6(a)]}}}
&  (A^\dag \ox B^\dag)^\dag \ar[d]_{\lambda_\pr^{-1}} \ar@{=}@/^4pc/[ddd]\\
(A\pr B)^\dag \ar[r]^{\phi_{A\pr B}^{-1}}  \ar[ddr]_{\phi_{(A\pr B)^\dag}} 								  \ar@{}[dr]|{\mbox{\tiny { \bf [U.4]}}}
& A\pr B  \ar[r]^{\phi_A\pr\phi_B} \ar@{=}[d]
& A^\dag \pr B^\dag \ar[r]^{\phi_{A^\dag}\pr\phi_{B^\dag}}										 \ar@{}[d]|{\mbox{\tiny { \bf [U.4]}}\ \pr\mbox{\tiny { \bf [U.4]}}}
& (A^\dag)^\dag \pr (B^\dag)^\dag  \ar@{=}[d]\\
& A \pr B  \ar[rr]^{\iota \pr \iota} \ar[d]^{\iota}
& {}																					\ar@{}[d]|{\mbox{\tiny {\bf [$\dagger$-ldc.5(a)]}}}
& (A^\dag)^\dag \pr (B^\dag)^\dag  \ar[d]_{\lambda_\pr}\\
& ((A \pr B)^\dag)^\dag  \ar[rr]_{\lambda_\pr^\dag}
& {}
& (A^\dag \ox B^\dag)^\dag

* $\lambda_\oa$ is unitary because:

\[
\xymatrix{
A^\dag\pr B^\dag                            \ar[rrr]^{\phi_{A^\dag\pr B^\dag}} \ar[dr]^{\mx^{-1}}   \ar[ddd]_{\lambda_\pr} \ar@{}[dddr]|{\mbox{\tiny {\bf Lem. \ref{lemma:mixdagger}}}}
(A^\dag\pr B^\dag)^\dag               \ar@{}[dddl]|{\mbox{\tiny {\bf (Lem. \ref{lemma:mixdagger})}}^\dag} \\
& A^\dag\ox B^\dag                       \ar[r]^{\phi_{A^\dag\ox B^\dag}} \ar[d]_{\lambda_\ox}   \ar@{}[ur]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (vi)}}} \ar@{}[dr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (i)}}}
& (A^\dag\ox B^\dag)^\dag           \ar[ur]^{(\mx^{-1})^\dag}
& (A\pr B)^\dag                       \ar[r]^{\phi_{(A\pr B)^\dag}}  \ar@{}[dr]|{\mbox{\tiny {\bf Lems. \ref{lemma:cohUnitary} (vi), \ref{lemma:MUCProperties} (i)}}}
& ((A\pr B)^\dag)^\dag                \ar[u]_{\lambda_\ox^\dag} \ar[dr]^{((\mx^{-1})^\dag)^\dag}
(A\ox B)^\dag                            \ar[rrr]^{\phi_{(A\pr B)^\dag}} \ar[ur]^{(\mx^{-1})^\dag}
((A\ox B)^\dag)^\dag              \ar[uuu]_{\lambda_\pr^\dag}
\]

* $\lambda_\bot: \bot \rightarrow \top^\dagger$ is unitary because:

\[
\xymatrix{
\bot \ar[d]_{\lambda_\bot} \ar[rr]^{\varphi_\bot} & & \bot^\dagger \ar[d]^{(\lambda_\bot^{-1})^{\dagger}} \\
\top^\dagger \ar[urr]_{m^\dagger} \ar[rr]_{\varphi_{\top^\dagger} = (\varphi_{\top}^{-1})^\dagger} &  &\top^{\dagger \dagger}
\]

The left triangle commutes by [U.5(a)].  The right triangle commutes by [U.5(b)] and the functoriality of $\dag$.

* $\lambda_\top: \top \rightarrow \bot^\dagger$ is unitary because:

\[
\xymatrix{
\top \ar[d]_{\lambda_\top} \ar[rr]^{\varphi_\top} & & \top^\dagger \ar[d]^{(\lambda_\top^{-1})^{\dagger}} \\
\bot^\dagger \ar[urr]_{(m^{-1})^\dagger} \ar[rr]_{\varphi_{\bot^\dagger} = (\varphi_{\bot}^{-1})^\dagger} &  &\bot^{\dagger \dagger}
\]

The left triangle commutes by [U.5(a)].  The right triangle commutes by [U.5(b)] and the functoriality of $\dag$.

* $\varphi_A$ is unitary because the following square commutes by U.5(a).

\[
\xymatrix{
A \ar[r]^{\varphi_A} \ar[d]_{\varphi_A} & A^\dagger \ar[d]^{(\varphi^{-1})^\dagger} \\
A^\dagger \ar[r]^{\varphi_{A^\dagger}} & A^{\dagger \dagger}
\]

* $m: \bot \rightarrow \top$ is unitary because:

\[
\xymatrix{
\bot \ar[r]^{\varphi_\bot} \ar[d]_{\m} & \bot^\dagger \ar[d]^{(\m^{-1})^\dagger} \\
\top \ar[r]_{\varphi_\top} \ar[ur]^{\lambda_\top} & \top^\dagger
\]

The left and right triangles commute by [U.5(a)] and [U.5(b)] respectively. Hence, the outer squares commutes.

* $\mx_{A,B}: A \ox B \rightarrow A \oa B$ is unitary as:

\[ \xymatrix{
A \ox B<EMAIL_ADDRESS>\ox B}} \ar[r]^\mx<EMAIL_ADDRESS>\ox \varphi_B}  \ar@{}[drr]|{\mbox{\tiny {\bf nat.}}}   \ar@{}[urrr]|{\mbox{\tiny {\bf [U.3]}}}
& A \oa B \ar[r]^{\varphi_A \oa \varphi_B}
& A^\dag \oa B^\dag \ar[r]^{\lambda_\oa}
& (A \ox B)^\dag \\
& A^\dag \ox B^\dag \ar[u]^\mx \ar[dr]^{\lambda_\ox}   \ar@{}[ur]|{\mbox{\tiny {\bf Lem. \ref{lemma:mixdagger}}}} \\
A \oa B \ar[rrr] _{\varphi_{A\oa B}}    \ar@{}[urr]|{\mbox{\tiny {\bf [U.3]}}}
& (A \oa B)^\dag \ar[uu]_{\mx^\dag}
\]

* $\iota: A \rightarrow A^{\dagger \dagger}$ is unitary as in

\[
\xymatrix{
A \ar[d]_{\iota} \ar[r]^{\varphi_A} & A^\dagger \ar[d]^{(\iota^{-1})^\dagger} \ar[ld]_{\varphi_{A^\dagger}} \\
A^{\dagger \dagger} \ar[r]_{\varphi_{A^{\dagger \dagger}}} & A^{\dagger \dagger \dagger}
\]

the left triangle commutes by [U.4] and the right triangle commutes by:

\begin{eqnarray*}
(\iota^{-1})^\dagger &= & ((\varphi_{A^\dagger})^{-1} \varphi_A^{-1})^\dagger =  (((\varphi_{A}^{-1})^\dagger)^{-1} \varphi_A^{-1})^\dagger \\
& =  & ((\varphi_A^\dagger)(\varphi_A^{-1}) )^\dagger = ( \varphi_A^{-1} )^{\dagger} ( \varphi_A )^{\dagger \dagger} \\
& = &  \varphi_{A^\dagger} (\varphi_A)^{\dagger \dagger} = \varphi_{A^\dagger} (\varphi_{A^{\dagger \dagger}})
\end{eqnarray*}

* $a_\ox$ is unitary as:

\[
\xymatrix{
(A \ox B) \ox C                                           \ar[rrr]^{\varphi_{(A \ox B) \ox C}} \ar[ddddd]_{a_\ox} \ar[dr]_{\mx}   \ar@{}[drrr]|{\mbox{\tiny {\bf [U.3]}}}
& {}
& {}
& ( (A \ox B) \ox C )^\dagger                     \\ %1
& {}
(A \ox B) \pr C                                             \ar[r]^{\varphi_{A\ox B} \pr \varphi_{C}} \ar[d]_{\mx\pr 1}   \ar@{}[dr]|{\mbox{\tiny {\bf  [U.6]$\pr$(id)}}}
& (A \ox B)^\dag \pr C^\dag                               \ar[d]^{ \lambda_\oa^{-1}\pr 1}   \ar[ur]_{\lambda_\oa}   \ar@{}[dddr]|{\mbox{\tiny {\bf [\dag-ldc.1]}}}
& {}  \\ %2
& (A \pr B) \pr C                                            \ar[r]_{(\varphi_A \oa \varphi_B) \oa \varphi_C} \ar[d]_{a_\oa}  \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& (A^\dag \pr B^\dag) \pr C^\dag                         \ar[d]^{a_\oa}
& {} \\ %3
& A \pr (B \pr C)                                           \ar[r]_{\phi_A \pr (\phi_B \pr \phi_C)}   \ar@{}[dr]|{\mbox{\tiny {\bf (id)$\pr$[U.6] }}}
& A^\dag \pr (B^\dag \pr C^\dag)       \ar[d]^{1\oa\lambda_\oa}
& {} \\ %4
& A \pr (B \ox C)                                          \ar[r]_{\varphi_{A} \pr \varphi_{B\ox C}} \ar[u]^{1\pr \mx}
& A^\dag \pr (B \ox C)^\dag               \ar[dr]^{\lambda_\oa}
& {} \\ %5
A \ox (B \ox C)                                          \ar[rrr]^{\varphi_{A \ox (B \ox C)}} \ar[ur]^{\mx}   \ar@{}[urrr]|{\mbox{\tiny {\bf [U.3]}}}
& {}
& {}
& ( A \ox (B \ox C)  )^\dagger \ar[uuuuu]_{a_\ox^\dag} %6
\]

* $a_\oa$ is unitary because:

\[
\xymatrix{
(A \pr B) \pr C                                           \ar[rrr]^{\varphi_{(A \pr B) \pr C}} \ar[ddddd]_{a_\pr} \ar[dr]_{\mx^{-1}}   \ar@{}[drrr]|{\mbox{\tiny {\bf [U.3]}}}
& {}
& {}
& ( (A \pr B) \pr C )^\dagger                     \\ %1
& {}
(A \pr B) \ox C                                             \ar[r]^{\varphi_{A\pr B} \ox \varphi_{C}} \ar[d]_{\mx^{-1}\ox 1}   \ar@{}[dr]|{\mbox{\tiny {\bf  [U.6]$\ox$(id)}}}
& (A \pr B)^\dag \ox C^\dag                               \ar[d]^{ \lambda_\ox^{-1}\ox 1}   \ar[ur]_{\lambda_\pr}   \ar@{}[dddr]|{\mbox{\tiny {\bf [\dag-ldc.1]}}}
& {}  \\ %2
& (A \ox B) \ox C                                            \ar[r]_{(\varphi_A \pr \varphi_B) \pr \varphi_C} \ar[d]_{a_\ox}  \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& (A^\dag \ox B^\dag) \ox C^\dag                         \ar[d]^{a_\ox}
& {} \\ %3
& A \ox (B \ox C)                                           \ar[r]_{\phi_A \ox (\phi_B \ox \phi_C)}   \ar@{}[dr]|{\mbox{\tiny {\bf (id)$\ox$[U.6] }}}
& A^\dag \ox (B^\dag \ox C^\dag)       \ar[d]^{1\ox\lambda_\ox}
& {} \\ %4
& A \ox (B \pr C)                                          \ar[r]_{\varphi_{A} \ox \varphi_{B\pr C}} \ar[u]^{1\ox \mx^{-1}}
& A^\dag \ox (B \oa C)^\dag               \ar[dr]^{\lambda_\ox}
& {} \\ %5
A \oa (B \oa C)                                          \ar[rrr]^{\varphi_{A \oa (B \oa C)}} \ar[ur]^{\mx^{-1}}   \ar@{}[urrr]|{\mbox{\tiny {\bf [U.3]}}}
& {}
& {}
& ( A \oa (B \oa C)  )^\dagger \ar[uuuuu]_{a_\oa^\dag} %6
\]

* $c_\ox$ is unitary because:

\[
\xymatrix{
A\ox B                            \ar[rrr]^{\phi_{A\ox B}} \ar[dr]^{\mx} \ar[ddd]_{c_\ox}
& {}                                  \ar@{}[dr]|{\mbox{\tiny {\bf [U.6(b)]}}}
& {}
& (A\ox B)^\dag              \\
& A \pr B                        \ar[r]^{\phi_A \pr \phi_B} \ar[d]_{c_\pr}   \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& A^\dag \pr B^\dag       \ar[d]^{c_\pr} \ar[ur]^{\lambda_\pr}           \ar@{}[dr]|{\mbox{\tiny {\bf [$\dagger$-ldc.2(b)]}}}
& {} \\
& B \pr A                       \ar[r]^{\phi_B \pr \phi_A}
& B^\dag \pr A^\dag      \ar[dr]^{\lambda_\pr}
& {} \\
B\ox A                           \ar[ur]^{\mx} \ar[rrr]^{\phi_{B\ox A}}
& {}                                 \ar@{}[ur]|{\mbox{\tiny {\bf [U.6(b)]}}}
& {}
& (B\ox A)^\dag                 \ar[uuu]_{(c_\ox^{-1})^\dag = c_\ox^\dag}\\
\]

where the left square commutes commutes because

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\end{tikzpicture}

* $c_\pr$ is unitary because:

\[
\xymatrix{
A\pr B                            \ar[rrr]^{\phi_{A\pr B}}  \ar[dr]^{\mx^{-1}}   \ar[ddd]_{c_\pr}  \ar@{}[drrr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (vii)}}}
& (A\pr B)^\dag \\
& A \ox B                      \ar[r]^{\phi_{A\ox B}} \ar[d]_{c_\ox}    \ar@{}[dr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (xi)}}}
& (A\ox B)^\dag     \ar[ur]^{(\mx^{-1})^\dag} &\\
& B \ox A                      \ar[r]^{\phi_{B\ox A}}
& (B \ox A)^\dag    \ar[dr]^{(\mx^{-1})^\dag}  \ar[u]_{c_\ox^\dag} &\\
B\pr A                            \ar[rrr]^{\phi_{B\pr A}}   \ar[ur]^{\mx^{-1}}   \ar@{}[urrr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (vii)}}}
& (B\pr A)^\dag              \ar[uuu]_{c_\pr^\dag}
\]

where the left square commutes for the same reason and the right square is the dagger of the left square.

* $\partial_L$ is unitary because:

\xymatrix{
& {}
& {}
& {}
& {}
& {}
& {}
& {}
& {}\\
& {}
& {}
& {}
& {}
& {}
& {}
& {}
& {}\\
%                              0                                     %
& {}
& A\ox (B \pr C)                      \ar[rr]^{\phi_{A}\ox \phi_{B\pr C}}    \ar[d]_{\mx}       \ar@/^4pc/[rrrrrr]^{\phi_{A\ox (B\pr C)}} \ar@/_6pc/[dddddd]_{\partial_L}
& {}                                                                                \ar@{}[d]|{\mbox{\tiny {\bf [U.6(b)]}}}
& A^\dag \ox (B \pr C)^\dag            \ar@=[r]   \ar@=[d]                                                                \ar@{}[dr]|{\mbox{\tiny {\bf id}}}
& A^\dag \ox (B \pr C)^\dag            \ar[r]^{\mx}    \ar[d]^{1 \ox \lambda_\ox^{-1} }     \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}    \ar@{}[u]|{\mbox{\tiny {\bf [U6.(b)]}}}
& A^\dag \pr (B \pr C)^\dag            \ar[rr]^{\lambda_\pr}    \ar[d]^{1 \pr \lambda_\ox^{-1}}                                              \ar@{}[ddrr]|{\mbox{\tiny {\bf [$\dagger$-ldc.4(b)]}}}
& {}
& (A\ox (B\pr C))^\dag                         \\
%                              1                                     %
& {}
& A\pr (B \pr C)                      \ar[r]^{\phi_{A\pr (B \pr C)}}     \ar[d]_{a_\pr}                                \ar@{}[dr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (ix)}}}
& (A\pr (B\pr C))^\dag               \ar[r]^{\lambda_\ox^{-1}}      \ar[d]^{(a_\pr^{-1})^\dag}
& A^\dag \ox (B\pr C)^\dag           \ar[r]^{1\ox \lambda_\ox^{-1}}                                                 \ar@{}[d]|{\mbox{\tiny {\bf [$\dagger$-ldc.1(b)]}}}
& A^\dag \ox (B^\dag \ox C^\dag)     \ar[r]^{\mx }    \ar[d]^{a_\ox^{-1}}                                                 \ar@{}[dr]|{\mbox{\tiny {\bf mix}}}
& A^\dag \pr (B^\dag \ox C^\dag)             %   \ar[d]^{\partial_R}
& {}
& {}\\
%                              2                                     %
& {}
& (A\pr B)\pr C                       \ar[r]^{\phi_{(A\pr B)\pr C}}     \ar@=[d]
& ((A\pr B)\pr C)^\dag                \ar[r]^{\lambda_\ox^{-1}}                                                            \ar@{}[d]|{\mbox{\tiny {\bf [U.6(a)]}}}
& (A\pr B)^\dag \ox C^\dag           \ar[r]^{\lambda_\ox^{-1}\ox 1}    \ar[d]_{\mx}                             \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& (A^\dag \ox B^\dag) \ox C^\dag      \ar[r]^{ \mx\ox 1}     \ar[d]^{\mx}                                            \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& (A^\dag \pr B^\dag) \ox C^\dag                 \ar[d]^{\mx}\ar@=[r]          \ar[u]_{\partial_R}                                          \ar@{}[ddddr]|{\mbox{\tiny {\bf nat.}}}
& (A^\dag \pr B^\dag) \ox C^\dag                 \ar[dddd]^{\lambda_\pr\ox 1}
& {}\\
%                              3                                     %
& {}
& (A\pr B)\pr C                      \ar[rr]^{\phi_{A\pr B}\pr \phi_{C}}     \ar[d]_{\mx^{-1}\pr 1}                     \ar@{}[ll]|{\mbox{\tiny {\bf mix}}}     \ar@{}[drr]|{\mbox{\tiny{\bf (id)$\ox$(Lem. \ref{lemma:cohUnitary}   (vii))} } }
& {}
& (A\pr B)^\dag \pr C^\dag           \ar[r]^{\lambda_\ox^{-1}\pr 1}     \ar[d]_{\mx^\dag\pr 1}         \ar@{}[dr]|{\mbox{\tiny {\bf 1$\ox$(Lem. \ref{lemma:mixdagger})  }}}
& (A^\dag \ox B^\dag) \pr C^\dag                \ar[d]^{\mx\pr 1}  \ar[r]^{\mx\pr 1}                          \ar@{}[ddr]|{\mbox{\tiny {\bf id}}}
& (A^\dag \pr B^\dag) \pr C^\dag                \ar@=[dd]
& {}
& {}\\
%                              4                                     %
& {}
& (A\pr B)\ox C                       \ar[rr]^{\phi_{A\pr B}\ox \phi_{C}}    \ar@=[d]
& {}                                                                                \ar@{}[d]|{\mbox{\tiny {\bf [U.6(a)] }}}
& (A\pr B)^\dag \ox C^\dag           \ar[r]^{\lambda_\ox^{-1}\pr 1}     \ar[d]_{\lambda_\ox}            \ar@{}[dr]|{\mbox{\tiny {\bf id}}}
& (A^\dag\ox B^\dag) \ox C^\dag                \ar[d]^{\lambda_\ox\ox 1}
& {}
& {}
& {}\\
%                              5                                     %
& {}
& (A\pr B)\ox C                       \ar[r]^{\mx}     \ar@=[d]
& (A\pr B)\pr C                       \ar[r]^{\phi_{(A\pr B)\pr C}}                                                            \ar@{}[dr]|{\mbox{\tiny {\bf [U.6(a)]}}}
& ((A\pr B)\pr C)^\dag                \ar[r]^{\lambda_\ox^{-1}}
& (A\pr B)^\dag \ox C^\dag           \ar[r]^{ \lambda_\ox^{-1} \ox 1}     \ar@=[d]                               \ar@{}[dr]|{\mbox{\tiny {\bf id}}}
& (A^\dag \ox B^\dag) \ox C^\dag       \ar[d]^{\lambda_\ox \ox 1}
& {}
& {}\\
%                              6                                     %
& {}
& (A\pr B) \ox C                      \ar[rrr]^{\phi_{A\pr B}\ox\phi_{C}}                           \ar@/_4pc/[rrrrrr]_{\phi_{(A\pr B) \ox C}}
& {}
& {}
& (A \pr B)^\dag \ox C^\dag          \ar@=[r]                                                                                \ar@{}[d]|{\mbox{\tiny {\bf [U.6(a)]}}}
& (A \pr B)^\dag \ox C^\dag          \ar[r]^{\mx}
& (A \pr B)^\dag \pr C^\dag          \ar[r]^{\lambda_\pr}
&  ((A \pr B) \ox C)^\dag               \ar[uuuuuu]^{\partial_L^\dag}\\
& {}
& {}
& {}
& {}
& {}
& {}
& {}
& {}\\

* $\partial_R$ is unitary because:

\[ \hspace{-1.25cm}
\xymatrix{
& {}
& {}
& {}
& {}
& {}\\
(\!A\!\pr\! B\!) \!\ox\! C                           \ar[r]^{\mx} \ar[d]_{\partial_R}    \ar@/^4pc/[rrrrr]^{\phi_{(A\pr B) \ox C }}
& (\!A\!\pr\! B\!) \!\pr\! C                      \ar[r]^{\mx^{-1}\pr 1}                                                        \ar@{}[d]|{\mbox{\tiny {\bf [\dag-ldc.4]}}}
& (\!A\!\ox\! B\!) \!\pr\! C                      \ar[r]^{\phi_{(A\ox B) \pr C}}                                              \ar@{}[dr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary}  (xiii)}}}  \ar@{}[ur]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (vii), \ref{lemma:MUCProperties}}}}
& (\!(\!A\!\ox\! B\!) \!\pr\! C \!)^\dag         \ar[r]^{(\mx^{-1}\pr 1 )^\dag} \ar[d]_{\partial_L^\dag}
& (\!(\!A\!\pr\! B\!) \!\pr\! C \!)^\dag         \ar[r]^{\mx^\dag}                                                               \ar@{}[d]|{\mbox{\tiny {\bf [\dag-ldc.4]}}^\dag}
& (\!(\!A\!\pr\! B\!) \!\ox\! C \!)^\dag \\
A\!\pr\! (\!B \!\ox\! C\!)                           \ar[r]^{\mx^{-1}}   \ar@/_4pc/[rrrrr]_{\phi_{A\pr (B \ox C) }}
& A\!\pr\! (\!B \!\pr\! C\!)                      \ar[r]^{1 \pr \mx}
& A\!\ox\! (\!B \!\pr\! C\!)                      \ar[r]^{\phi_{A\ox (B \pr C)}} \ar[u]_{\partial_L}  \ar@{}[dr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary}  (vii), \ref{lemma:MUCProperties}}}}
& (\!A\!\ox\! (\!B\!\pr\! C\!) \!)^\dag         \ar[r]^{(\mx \pr 1 )^\dag}
& (\!A\!\pr\! (\!B \!\pr\! C\!) \!)^\dag         \ar[r]^{(\mx^{-1})^\dag}
& (\!A\!\pr\! (\!B \!\ox\! C\!) \!)^\dag        \ar[u]_{\partial_R^\dag}\\
& {}
& {}
& {}
& {}
& {}
\]

§.§ Unitary categories

In this section, we provide a general construction of a unitary category from a $\dagger$-isomix category.  This provides a general and important technique of constructing unitary categories.   The construction is based on identifying objects with pre-unitary structure: the tensor units always have a canonical pre-unitary structure so the construction always produces a non-empty category.   However, to ensure that an application of the construction yields a unitary category in which there are object which are not isomorphic to the units, one must exhibit concretely such objects.  Fortunately this is often not difficult to do, making the construction quite useful.

In a $\dagger$-mix category a pre-unitary object is an object $U$, which in the core, together with an isomorphism $\alpha: U \to U^\dagger$ such that  $\alpha (\alpha^{-1})^\dagger = \iota$.

Suppose $\X$ is a $\dagger$-mix category, then define ${\sf Unitary}(\X)$, the unitary core of $\X$, as follows:

Objects: Pre-unitary objects $(U, \alpha)$,

Maps: $(U, \alpha) \to^f (V, \beta)$ where $U \to^f V $ is in $\X$.

For and $\dagger$-isomix category the unitary core, $\Unitary(\X)$, is a compact $\dagger$-LDC with tensor and par defined by

\[ (\top,{\sf m}^{-1}\lambda_\bot: \top \to \top^\dagger) ~~~~~(A, \alpha) \ox (B, \beta) := (A \ox B, \mx^{-1}(\alpha \oa \beta) \lambda_\oa: A \ox B \to (A \ox B)^\dagger)\]

\[ (\bot, {\sf m} ~\lambda_\top: \bot \to \bot^\dagger)  ~~~~~(A, \alpha) \oa (B, \beta) := (A \oa B, \mx(\alpha \ox \beta) \lambda_\ox: A \oa B \to (A \oa B)^\dagger) \]

and $(U,\alpha)^\dagger := (U^\dagger, (\alpha^{-1})^\dagger)$.

The proof uses the techniques of Lemma <ref>.

Note that, as the map and tensor structure is inherited, it suffices to show that these objects are all pre-unitary objects.  Starting with $(U \alpha)^\dagger$
we have:

\[ (\alpha^{-1})^\dagger (((\alpha^{-1})^\dagger)^{-1})^\dagger = (\alpha^{-1})^\dagger (\alpha^\dagger)^\dagger = (\alpha^\dagger \alpha^{-1})^\dagger = (\iota^{-1})^\dagger = \iota \]

For the tensor and par we have:

\begin{eqnarray*}
{\sf m}^{-1}\lambda_\bot (({\sf m}^{-1}\lambda_\bot)^{-1})^\dagger & = & {\sf m}^{-1}\lambda_\bot {\sf m}^\dagger \lambda_\bot^\dagger \\
& = &  {\sf m}^{-1} {\sf m} \lambda_\top  \lambda_\bot^\dagger  = \iota \\
\mx^{-1}(\alpha \oa \beta) \lambda_\oa ((\mx^{-1}(\alpha \oa \beta) \lambda_\oa)^{-1})^\dagger
& = & \mx^{-1}(\alpha \oa \beta) \lambda_\oa (\mx^\dagger) (\alpha^{-1} \oa \beta^{-1})^\dagger (\lambda_\oa^{-1})^\dagger \\
& = & \mx^{-1}(\alpha \oa \beta) \mx \lambda_\ox (\alpha^{-1} \oa \beta^{-1})^\dagger (\lambda_\oa^{-1})^\dagger \\
& = & (\alpha \ox \beta) \lambda_\ox (\alpha^{-1} \oa \beta^{-1})^\dagger (\lambda_\oa^{-1})^\dagger \\
& = & (\alpha \oa \beta) ((\alpha^{-1})^\dagger \oa (\alpha^{-1})^\dagger) \lambda_\ox (\lambda_\oa^{-1})^\dagger \\
& = & (\iota \oa \iota) \lambda_\ox (\lambda_\oa^{-1})^\dagger  = \iota \\
{\sf m} \lambda_\top (({\sf m} \lambda_\top)^{-1})^\dagger & = & {\sf m} \lambda_\top ({\sf m}^{-1})^\dagger  (\lambda_\top^{-1})^\dagger  \\
& = & {\sf m} ~{\sf m}^{-1} \lambda_\bot (\lambda_\top^{-1})^\dagger = \iota \\
\mx(\alpha \ox \beta) \lambda_\ox ((\mx(\alpha \ox \beta) \lambda_\ox)^{-1})^\dagger
& = & \mx(\alpha \ox \beta) \lambda_\ox (\mx^{-1})^\dagger (\alpha^{-1} \ox \beta^{-1})^\dagger (\lambda_\ox^{-1})^\dagger \\
& = & (\alpha \oa \beta) \mx~ \mx^{-1} \lambda_\oa (\alpha^{-1} \ox \beta^{-1})^\dagger (\lambda_\ox^{-1})^\dagger \\
&= & (\alpha \oa \beta) ((\alpha^{-1})^\dagger \oa (\beta^{-1})^\dagger) \lambda_\oa (\lambda_\ox^{-1})^\dagger \\
& = & (\iota \oa \iota) \lambda_\oa (\lambda_\ox^{-1})^\dagger  = \iota.
\end{eqnarray*}

This makes $\Unitary(\X)$ into a compact $\dagger$-isomix category with all the structure inherited directly from $\X$.
However, more is true: each object now has an obvious unitary structure.  This gives:

If $\X$ is any $\dagger$-isomix category, then $\Unitary(\X)$ is a unitary category with a full and faithful underlying $\dagger$-isomix functor to $U: {\sf Unitary}(\X) \to \X$.

The laxors are all identity maps so that the underlying functors is immediately a $\dagger$-mix functor.

It remains to show that every object is unitary:  we set the unitary structure of an object to be $\alpha: (X,\alpha) \to (X,\alpha)^\dagger$.   However, [U.1] – [U.6] are immediately satisfied by construction implying this provides unitary structure for every object.

A unitary category being a compact LDC is linearly equivalent, using ${\sf Mx}^*_\uparrow: (\X, \oa,\oa) \to (\X,\ox,\oa)$ (see Corollary <ref>) to the underlying monoidal category based on the par using the mix functor.   We now show that for a unitary category one can induce a stationary on objects dagger on $(\X,\oa,\oa)$.  We denote this dagger by
$(\_)^\ddagger$ and define it by $f^\ddagger := \varphi_Bf^\dagger\varphi_A^{-1}$ as illustrated by the left diagram below:

\[ \xymatrix{ B \ar[d]_{\varphi_B}\ar[rr]^{f^\ddagger} \ar@{}[rrd]|{:=}& & A \ar[d]^{\varphi_A} \\
B^\dagger \ar[rr]_{f^\dagger} && A^\dagger} ~~~~~~~~~~~~~
\xymatrix{ A \ar@/_1pc/[dd]_{\iota} \ar[d]^{\varphi_A}\ar[rr]^{f^{\ddagger\ddagger}} & & B \ar[d]_{\varphi_B} \ar@/^1pc/[dd]^{\iota} \\
A^\dagger \ar[d]^{(\varphi_A^{-1})^\dagger} \ar[rr]_{(f^\ddagger)^\dagger} && B^\dagger \ar[d]_{(\varphi_B^{-1})^\dagger} \\
A^{\dagger\dagger} \ar[rr]_{f^{\dagger\dagger}} & & B^{\dagger\dagger} } \]

This new dagger clearly preserves composition and is also a stationary on objects involution as proven by the second diagram:
the lower square of this diagram is the dagger of the inverted definition and the resulting outer square is the naturality of $\iota$ forcing $f^{\ddagger\ddagger} = f$.

We next observe that if $u: X \to Y$ is unitary in $\X$ if and only if $u^{-1}= u^\ddagger$.  This makes unitary in the tradition sense of categorical quantum mechanics coincide
with the notion introduced here.   Thus, $u$ unitary in the sense here if and only if the diagram below commutes

\[ \xymatrix{ B \ar[d]_{\varphi_B} \ar[rr]^{u^{-1}} && A  \ar[d]^{\varphi_A} \\ B^\dagger \ar[rr]_{u^\dagger} && A} \]

but this diagram commutes if and only if $u^{-1} = u^\ddagger$.

A $\dagger$-Frobenius functor, $F: \X \to \Y$, between isomix categories with unitary structure preserves unitary structure in case whenever $\varphi_A$ is unitary in $\X$ then
$F(\varphi_A)\rho^F$ is unitary structure in $\Y$.

We now show that ${\sf Mx}_\downarrow: (\X,\oa,\oa) \to (\X,\ox,\oa)$ provides a unitary structure preserving equivalence of a dagger monoidal category into a unitary category:

Unitary categories are $\dagger$-linearly equivalent via the mix functor ${\sf Mx}_\downarrow: (\X,\oa,\oa) \to (\X,\ox,\oa)$ to the underlying dagger monoidal category on the par.
Furthermore, closed unitary categories under this equivalence become dagger compact closed categories.

We must exhibit a preservator, that is a natural transformation showing that the involution is preserved:

\[ \infer={A \to_{\varphi_A} A^\dagger}{{\sf Mx}_\downarrow(A^\ddagger) \to^{\varphi_A} {\sf Mx}_\downarrow(A)^\dagger} \]

Note that $\varphi$ is a natural transformation by the definition of $(\_)^\ddagger$ and its coherence requirements
make it a linear natural equivalence.  Making this the preservator immediately means that unitary structure is preserved.

Finally we must show that unitary linear duals under ${\sf Mx}^{*}_\downarrow$ become $\ddagger$-duals.  Given $(\eta,\epsilon): A \dashvv_u B$  we must
show that under ${\sf Mx}^{*}_\downarrow$ this produces a dagger dual.  ${\sf Mx}^{*}_\downarrow(\eta) = {\sf m} ~\eta: \bot \to A \oa B$ and
${\sf Mx}^{*}_\downarrow(\epsilon) = {\sf mx}^{-1} \epsilon: B \oa A \to \bot$
We then require that $c_\oa {\sf Mx}^{*}_\downarrow(\epsilon) = {\sf Mx}^{*}_\downarrow(\eta)^\ddagger$. This is provided by:

\[ \xymatrix{A \oa B  \ar[d]^{{\sf mx}^{-1}}  \ar@/_2pc/[ddd]_{\varphi_{A \oa B}} \ar[r]^{c_\oa} & B \oa A \ar@/^1pc/[rr]^{{\sf Mx}^{*}_\downarrow(\epsilon)} \ar[r]_{{\sf mx}^{-1}}
& A \ox B \ar[r]_{\epsilon} & \bot \ar[dd]_{{\sf m}} \ar@/^2pc/[ddd]^{\varphi_\bot} \ar@/_/[dddl]_{\lambda_\bot}\\
A \ox B \ar[d]^{\varphi_A \ox \varphi_B} \ar@/_/[rru]^{c_\ox} \\
A^\dagger \ox B^\dagger \ar[d]^{\lambda_\ox} & & & \top \ar[d]_{\lambda_\top} \\
(A \oa )^\dagger \ar@{}[rrruuu]|{{\rm Defn.} ~\ref{unitary-duals}~(b)} \ar@/_2pc/[rrr]_{{\sf Mx}^{*}_\downarrow(\eta)^\dagger} \ar[rr]_{\eta^\dagger} & & \top^\dagger \ar[r]_{{\sf m}^\dagger} & \bot^\dagger } \]

§.§ Mixed unitary categories

A mixed unitary category (MUC) is a strong isomix Frobenius functor $M: \U \to \X$ from a unitary category to a $\dagger$-isomix category, $\X$, such that $M$ factors through the core of $\X$.
We already have various sources of examples of MUCs.  The inclusions of the full subcategory of unitary objects of any $\dagger$-isomix category with unitary structure is a MUC.   More interestingly, for any $\dagger$-isomix category, $\X$, $U: {\sf Unitary}(\X) \to \X$ is  a MUC.

Our main objective in this section is to explain the couniversal property of the unitary construction.

Mix unitary categories organize themselves into a 2-category ${\sf MUC}$ (although we shall not discuss the 2-cell structure):

0-cells:  Are mix unitary categories $M: \U \to \X$;

1-cells:  Are MUC morphisms: these are squares of $\dagger$-mix Frobenius functors $(F',F,\gamma): M \to N$ commuting up to a $\dagger$-linear natural isomorphism $\gamma$:

\[ \xymatrix{ \U \ar[d]_{F'} \ar@{}[drr]|{\Downarrow~\gamma} \ar[rr]^M & & \X \ar[d]^F \\ \V \ar[rr]_{N} & & \Y} \]

The functor $F': \U \to \V$ is between unitary categories and we demand of it that preserves unitary structure in the
sense of Definition <ref>, thus, whenever ${\varphi_A}$ is the unitary structure  then $F(\varphi_A) \rho^F$ is unitary structure.

2-cells: These are “pillows”  of natural transformations.

We remark that we have observed that any MUC can be “simplified” to a dagger monoidal category with a strong $\dagger$-mix Frobenius functor into a $\dagger$–isomix category: this is achieved by precomposing with ${\sf Mx}_\downarrow$.   This may seem a worthwhile simplification but it should be recognized that it simply transfers complexity from the unitary category itself onto the preservator which must now “create” unitary structure:

\[ \xymatrix{\U  \ar[d]_{{\sf Mx}^{*}_\downarrow} \ar[rr]^M & & \C \ar@{=}[d] \\
\U_\downarrow \ar[rr]_{{\sf Mx}_\downarrow;M} & & \C} \]

Here $\U_\downarrow = (\U,\oa,\oa)$ is viewed as a dagger monoidal category and ${\sf Mx}_\downarrow^{*}$ is the inverse of ${\sf Mx}_\downarrow$.
The point is that the preservator of the lower arrow ${\sf Mx}_\downarrow;M$ is non-trivial as it must encode the unitary structure of $\U$.

Our objective is now to show that the unitary construction of the previous section gives rise to a right adjoint to the underlying 2-functor $U: {\sf MUC} \to {\sf MCC}$  where
the 2-category ${\sf MCC}$ is defined as:

0-cells: Its objects are  mix core categories, that is strong $\dagger$-Frobenius functors $V: \C \to \Y$ where $\C$ is a compact $\dagger$-LDC, $\Y$ is a $\dagger$-isomix category,
and $V$ factors through the core of $\Y$.  An example of a mix core category is, of course, the inclusion of the core into a $\dagger$-isomix category $C:{\sf Core}(\X) \to \X$;

1-cells:   The 1-cells are squares of mix Frobenius functors which commute up to a linear natural isomorphism;

2-cells:    Are pillows of natural transformations (which we shall ignore).

Preliminary to our next result we prove that Frobenius functors preserve pre-unitary objects:

Frobenius functors between compact LDCs preserve preunitary objects.  Thus, suppose $F: \X \to \Y$ is a Frobenius $\dagger$-functor between compact $\dagger$-LDCs, then if $(A,\varphi)$ is a preunitary object and then $(F(A),F(\varphi)\rho)$ is a preunitary object in $\Y$.

To prove that $(F_\ox(A)F(\varphi)\rho)$ is a preunitary objects, one has the following computation:

\begin{eqnarray*}
F(\varphi)\rho ((F(\varphi) \rho)^{-1})^\dagger
& = & F(\varphi)\rho (F(\varphi^{-1})^\dagger (\rho^{-1})^\dagger \\
& = & F(\varphi (\varphi^{-1})^\dagger) \rho (\rho^{-1})^\dagger \\
& = & F(\iota) \rho (\rho^{-1})^\dagger = \iota.
\end{eqnarray*}

$U: {\sf MUC} \to {\sf MCC}$ has a right adjoint ${\sf Unitary}: {\sf MCC} \to {\sf MUC}; [\C \to^C \X] \mapsto [{\sf Unitary}(\C) \to^{U;C} \X]$.

The couniversal diagram is as follows:

\[ \xymatrix{ {\U \to^M \X} \ar[rr]^{(F,G,\gamma)} \ar[d]_{(F^\flat,G,\gamma^\flat)} && {\C \to^V \Y} \\
{{\sf Unitary}(\C) \to \Y} \ar[urr]_{\epsilon} } \]

where $\epsilon$ is the square on the left and $(F^\flat,G,\gamma^\flat)$ is the square on the right:

\[ \xymatrix{{\sf Unitary}(\C) \ar[d]_U \ar[r]^{~~~~U} & \C \ar[r]^V & \Y \ar@{=}[d] \\
\C \ar[rr]_V && \Y}
\xymatrix{\U \ar[dr]_F \ar[d]^{F^\flat} \ar@{}[drr]|{~~~~~~~~~\uparrow~\gamma} \ar[rr]^M & & \X \ar[d]^{G} \\
{\sf Unitary}(\C) \ar[r]_{~~~~U} & \C \ar[r]_V & \Y} \]

It remains only to explain the triangle in the second square $F$ is a Fobenius functor and so preserves pre-unitary structure.
This means that each $(U,\varphi_U)$ in $\U$ is carried by $F$ onto a pre-unitary object in $\C$, $(F(U),F(\varphi)\rho^F)$.  But a pre-unitary object in
$\C$ is an object of ${\sf Unitary}(\C)$ and this determines $F^\flat$.   The functor is uniquely determined as it must preserve the unitary structure.

This proposition means that in building a MUC from a $\dagger$-LDC it suffices to show that the core contains non-trivial pre-unitary objects.  We shall use this fact below.

§ EXAMPLES OF MIXED UNITARY CATEGORIES

In this section we present a number of examples of MUCs.  We have already noted that dagger monoidal categories are automatically unitary categories in which the unitary
structure is given by identity maps.  The identity functors then gives a rather trivial MUC.  More excitingly – and beyond the scope of these notes (although we hope to return to
describe this example in more detail later) – one can take the bicompletion of the $\dagger$-monoidal category: this is a non-trivial $\dagger$-isomix $*$-autonomous category
extension of the original $\dagger$-monoidal category and provides, thus, an interesting example of how MUCs arise.

Our purpose in this section is to exhibit some non-trivial manifestations of the various structural components of a MUC.  To this end we discuss in some detail three basic examples.

§.§ Finite dimensional framed vector spaces

Our first example is significant as it provides an example of a unitary category which is non-trivial (in the sense that the
unitary structure is not given by identity maps).  It is the category of “framed” finite dimensional vector spaces, where a frame in this context is just a
choice of basis.  Thus the objects in this category are vector spaces with a chosen basis while the maps, ignoring the basis, are simply homomorphisms of
the vector spaces.

The category of finite dimensional framed vectors spaces, ${\sf FFVec}_K$, is a monoidal category defined as follows:

Objects:  The objects are pairs $(V,{\cal V})$ where $V$ is a finite dimensional $K$-vector space and ${\cal V} = \{ v_1,...,v_n \}$ is a basis.

Maps:  A map $(V,{\cal V}) \xrightarrow{f} (W,{\cal W})$ is a linear map  $V \xrightarrow{f} W$ in ${\sf FdVec}_K$.

Tensor: $(V,{\cal V}) \ox (W,{\cal W}) = (V \ox W,\{ v \ox w | v \in {\cal V}, w \in {\cal W} \})$ where $V \ox W$ is the usual tensor product.  The unit is
$(K,\{ e \})$ where $e$ is the unit of the field $K$.

To define the “dagger” we  must first choose an involution $\overline{(\_)}: K \to K$, that is a field homomorphism with $k = \overline{(\overline{k})}$. The
canonical example being conjugation of the complex numbers, however, the involution can be arbitrarily chosen – so could also, for example, be the identity.
This involution then can be extended to a (covariant) functor:

\[ \overline{(\_)}: {\sf FFVec}_K \to {\sf FFVec}_K; \begin{array}[c]{c} \xymatrix{(V,{\cal V}) \ar[d]^{f} \\ (W,{\cal W})} \end{array}
\mapsto \begin{array}[c]{c} \xymatrix{\overline{(V,{\cal V}) } \ar[d]^{\overline{f}} \\ \overline{(W,{\cal W}) }} \end{array} \]

where $\overline{(V,{\cal V})}$ is the vector space with the same basis but with the conjugate action $c~\overline{\cdot}~v = \overline{c} \cdot  v$.  The conjugate
homomorphism, $\overline{f}$, is then the same underlying map which is homomorphism between the conjugate spaces.

${\sf FFVec}_K$ is also a compact closed category with $(V,{\cal B})^{*} = (V^{*}, \{ \widetilde{b_i} | b_i \in {\cal B} \})$ where

\[ V^{*} = V \lollipop K~~~~\mbox{and}~~~~\widetilde{b_i}: V \to K; \left(\sum_j \beta_j \cdot b_j \right) \mapsto  \beta_i \]

This makes $(\_)^{*}: {\sf FFVec}_K^{\rm op} \to {\sf FFVec}_K$ a contravariant functor whose action is determined by precomposition.   Finally we
define the “dagger” to be the composite $(V,{\cal B})^\dagger = \overline{(V,{\cal B})^{*}}$.

This is a compact LDC with tensor and par being identified (so the linear distribution is the associator) and is isomix.  We must show that it is a $\dagger$-LDC.
Towards this aim we define the required natural transformations on the basis:

\[ \lambda_\ox = \lambda_\oa: (V,{\cal V})^\dag \ox (W,{\cal W})^\dag \to ((V,{\cal V}) \ox (W,{\cal W}))^\dag;  \widetilde{v_i} \ox \widetilde{w_j} \mapsto \widetilde{v_i \ox w_j} \]

\[ \lambda_\top = \lambda_\bot: (K,\{ e\}) \to (K,\{ e\})^\dag; k \mapsto \overline{k} \]

\[ \iota: (V,{\cal V}) \to ((V,{\cal V})^\dag)^\dag; v \mapsto \lambda f. f(v) \]

Note that the last transformation is given in a basis independent manner. Importantly, it may also be given in a basis dependent manner as
$\iota(v_i) = \widetilde{\widetilde{v_i}}$ as the behaviour of these two maps is the  same when applied to the basis of $(V,{\cal V})^\dag$ namely
the elements $\widetilde{v_j}$:

\[ \iota(v_i)(\widetilde{v_j}) = (\lambda f. f(v_i)) \widetilde{v_j} = \widetilde{v_j} v_i = \delta_{i,j} = \widetilde{\widetilde{v_i}}(\widetilde{v_j})  \]

Also note that $\widetilde{v_i \ox w_j} = (\widetilde{v_i} \ox \widetilde{w_j}) u_\ox$, where $u_\ox: K \ox K \to K$ is the multiplication of the field.
With these definitions in hand it is straightforward to check that this gives a $\dagger$-LDC by checking the required coherences on basis elements.
To demonstrate the technique consider the coherence [$\dagger$-ldc.4]:

\[ \xymatrix{A \oa B \ar[d]_{\iota \oa \iota} \ar[rr]^\iota & & ((A \oa B)^\dag)^\dag \ar[d]^{\lambda_\ox^\dag} \\
(A^\dag)^\dag \oa (B^\dag)^\dag  \ar[rr]_{\lambda_\oa} & & (A^\dag \ox B^\dag)^\dag} \]

We must show (identifying tensor and par) that $\lambda_\ox^\dag (\iota(a_i \ox b_j)) = \lambda_\ox(\iota \ox \iota(a_i \ox b_j))$.  Now the result  is a
higher-order term so it suffices to show the evaluations on basis elements are the same.  This means we need to show:
$\lambda_\ox^\dag (\iota(a_i \ox b_j))(\widetilde{a_p} \ox \widetilde{b_q}) = \lambda_\ox(\iota \ox \iota(a_i \ox b_j))(\widetilde{a_p} \ox \widetilde{b_q})$

\begin{eqnarray*}
(\lambda_\ox(\iota \ox \iota(a_i \ox b_j)))(\widetilde{a_p} \ox \widetilde{b_q}) & = & (\lambda_\ox(\widetilde{\widetilde{a_i}} \ox \widetilde{\widetilde{b_j}}))(\widetilde{a_p} \ox \widetilde{b_q}) \\
&  = & (\widetilde{\widetilde{a_i} \ox \widetilde{b_j}}) (\widetilde{a_p} \ox \widetilde{b_q}) \\
& = & (\widetilde{a_p} \ox \widetilde{b_q})(\widetilde{\widetilde{a_i}} \ox \widetilde{\widetilde{b_j}}) u_\ox ~~~\mbox{(diagrammatic order)}\\
& = & \delta_{p,i} \delta_{q,j} \\
(\lambda_\ox^\dag (\iota(a_i \ox b_j)))(\widetilde{a_p} \ox \widetilde{b_q})
& = & (\lambda_\ox^\dag (\widetilde{\widetilde{a_i \ox b_j}}))(\widetilde{a_p} \ox \widetilde{b_q})  \\
& = & (\widetilde{a_p} \ox \widetilde{b_q}) \lambda_\ox \widetilde{\widetilde{a_i \ox b_j}} ~~~\mbox{(diagrammatic order)}\\
& = & \widetilde{a_p \ox b_q} \widetilde{\widetilde{a_i \ox b_j}} \\
& = & \delta_{p,i} \delta_{q,j}
\end{eqnarray*}

The unitary structure for this example is defined by:

\[ \varphi_{(V,{\cal  V})}: (V,{\cal  V}) \to (V,{\cal  V})^\dag; v_i \mapsto \widetilde{v_i} \]

and it remains to check the coherences [U.3]–[U.6].  First note that [U.4] holds immediately by the observation above that
$\iota(v_i) = \widetilde{\widetilde{v_i}}$.  For [U.3] we require that $\varphi_{A^\dag}(\widetilde{a_i}) = (\varphi_A^{-1})^\dag)(\widetilde{a_i})$
the result is a higher-order term so we may check that the evaluations are the same on basis elements:

\begin{eqnarray*}
(\varphi_{A^\dag}(\widetilde{a_i}) ) (\widetilde{a_j}) & = & \widetilde{\widetilde{a_i}}(\widetilde{a_j}) = \delta_{i,j} \\
((\varphi_{A}^{-1})^\dag(\widetilde{a_i}))(\widetilde{a_j}) & = & \widetilde{a_i} (\varphi_{A}^{-1}(\widetilde{a_j})) = \widetilde{a_i}(a_j) =  \delta_{i,j}
\end{eqnarray*}

Note that [U.5](a) and [U.5](b), in this example, requires $\lambda_\top = \varphi_\top$ which can easily be verified as each reduces to conjugation.
[U.6](a) and [U.6](b), in this example, are the same requirement which is verified by:

\[ \lambda_\ox(\varphi_A \ox \varphi_B(a_i \ox b_j) ) = \lambda_\ox (\widetilde{a_i} \ox \widetilde{b_j}) = \widetilde{a_i \ox b_j} = \varphi_{A \ox B} (a_i \ox b_j) \]

This gives:

${\sf FFVec}_K$ with the unitary structure above is a MUC.

This raises the question of what precisely the unitary maps of this example are.  To elucidate this we note that  a functor can easily be constructed
$U:{\sf FFVec}_K \to {\sf Mat}(K)$ where, for each object in ${\sf FFVec}_K$ we choose a total order on the elements of the basis and note that
any map is then given by a matrix acting on the bases: thus a matrix in ${\sf Mat}(K)$ with the appropriate dimensions.  We now observe:

An isomorphism $u: (A,{\cal A}) \to (B,{\cal B})$ in ${\sf FFVec}_K$ is unitary if and only if $U(f)$ is unitary in ${\sf Mat}(\X)$.

While $U$ does not preserve $(\_)^\dag$ on the nose it does so up to the natural equivalence determined by $U(\varphi_A)$ which being a basis
permutation is a unitary equivalence.   Thus, it is not hard to see that the following diagram commutes:

\[ \xymatrix{ U(B,{\cal B})  \ar[d]_{U(f)^\dag} \ar[rr]^{U(\varphi_B)} & &U((B,{\cal B})^\dag) \ar[d]^{U(f^\dag)} \\
U(A,{\cal A}) \ar[rr]_{U(\varphi_A)} & & U((A,{\cal A})^\dag) } \]

where recall in the category of matrices dagger is stationary on objects so $U(B,{\cal B}) = U(B,{\cal B})^\dag$.

Now suppose $u$ is unitary in ${\sf FFVec}_K$  then $u^{-1} = \varphi_B u^\dagger \varphi_A^{-1}$ so that

\[ U(u)^{-1} = U(u^{-1}) = U(\varphi_B u^\dagger \varphi_A^{-1}) = U(\varphi_B) U(u^\dagger) U(\varphi_A^{-1}) = U(u)^\dagger \]

so that its underlying map is unitary. Conversely, if $U(u)$ is unitary then

\[ U(u^{-1}) = U(u)^{-1} = U(u)^\dag = U(\varphi_B u^\dagger \varphi_A^{-1}) \]

which immediately implies, as $U$ is faithful, that $u$ is unitary in ${\sf FFVec}_K$.

One might reasonably regard this as a rather round about way to describe the standard notion of a unitary map.  However, two things of importance have been
achieved.  First an example of a unitary category with a non-stationary dagger and, thus, a non-identity unitary structure, has been exhibited.  Second we have
shown how the standard unitary structure may be re-expressed in this formalism using straightforward non-stationary constructs.

§.§ Finiteness spaces

Finiteness spaces were introduced by [15] as a model of linear logic.  The type system can then be used to produce a typed system for infinite dimensional matrix multiplication in which no sums become infinite.   The system of infinite dimensional matrices forms an isomix $*$-autonomous category.  If one take matrices over the complex numbers then there is a natural notion of conjugation and this, in turn, gives a $\dagger$-isomix category: by taking the unitary core one can then obtain a MUC.

Let $X$ be a set and $\mathcal{F}$ be a subset of $\mathcal{P} (X)$.  Define the set,

\[\mathcal{F}^\perp:=\{ x' \subseteq X |\forall x \in \mathcal{F},  x\cap x'\text{ is finite} \}\]

A finiteness space is a pair $\X:=(X,\mathcal{F})$ so that $\mathcal{F}^{\perp\perp} = \mathcal{F}$.    $X$ is called the web, and the elements of $\mathcal{F}$ are called the finitary sets of the finiteness space.

Observe that if the web, $X$, is finite, then $\mathcal{F}$ is forced to be the whole powerset of $X$.  Finiteness spaces organize themselves into a (symmetric) $*$-autonomous category:

Objects:  Finiteness spaces $\X:=(X, \mathcal{F})$.

Maps:  A map $R:(X, \mathcal{F}) \to (Y, \mathcal{G})$ is a relation $R:X\to Y$ so that:

* For all $x \in \mathcal{F}$, $xR \in \mathcal{G}$.

* For all $y \in \mathcal{G}^\perp$, $Ry \in \mathcal{F}^\perp$

Composition and identities:  same as in sets and relations.

Monoidal tensor:  Given maps $R:\X_1\to\Y_1$ and $S:\X_2\to\Y_2$:

\[R\ox S := \{((x_1, x_2), (y_1,y_2))| (x_1,y_1)\in R, (x_2,y_2) \in S \}\]

Monoidal tensor unit:  $\top := (\{*\}, \mathcal{P}(\{*\}))$

Dualizing functor:  $(X, \mathcal{F})^* := (X, \mathcal{F}^\perp)$

Finiteness spaces form an isomix $*$-autonomous category, where the core is the full subcategory determined by objects whose webs are finite sets.

From the category of finiteness spaces we can build a category of finiteness matrices over the complex numbers, ${\sf FMat}(\C)$, its objects are
finiteness spaces and its maps are matrices whose support is a finiteness relation between the finiteness spaces.   This category is an isomix
$*$-auonomous category.  However, it also has a natural conjugation – essentially by taking conjugates of the matrices – thus, it has a natural dagger

The core of ${\sf FMat}(\C)$ is the subcategory of finite sets and this is a well-known $\dagger$-compact closed category and the inclusion
${\cal I}: {\sf Mat}(\C) \to {\sf FMat}(\C)$ provides an important example of a MUC.   Note, however, that the unitary structure is “trivial” in the
sense that it is given by identity maps.

§.§ Chu Spaces give MUCs

The Chu construction over a closed involutive monoidal category, which  has pullbacks, produces a $\dagger$-isomix LDC, ${\sf Chu}_\X(I)$, whose pre-unitary objects can be used, as  above, to form a mix unitary category.  To get the $*$-autonomous category and $\dagger$-structure on ${\sf Chu}_\X(I)$ we shall start by explaining how one can produce  involutive structure on the ${\sf Chu}$ category.  To achieve this we iteratively develop the structure of this category, starting with an involutive closed monoidal category, $\X$, which is not necessarily  symmetric.  Note that the fact that it is involutive means that it is both left and right closed which allows us to consider the non-commutative ${\sf Chu}$ construction: in this regard  we shall follow Jürgen Koslowski's construction [18] using simplified “Chu-cells” on the same dualizing object to obtain not a ${*}$-linear bicategory but a cyclic  $*$-autonomous category.  Furthermore, we shall choose a dualizing object which is involutive in order  to obtain an involutive cyclic $*$-autonomous  category.

Recall that an involutive object is an object $D$ of $\X$ with an isomorphism $d: \overline{D} \to D$ such that $\overline{d}d = \varepsilon$.
We can then define ${\sf Chu}_\X(D)$ as follows:

Objects: $(A, B, \psi_0, \psi_1)$ where $\psi_0: A \ox B \to D$ and $\psi_1: B \ox A \to D$  in $\X$ (these are the simplified Chu cells).

Arrows: $(f,g): (A, B, \psi_0,\psi_1) \to (A', B', \psi_0',\psi_1')$ where $f: A \to A'$ and $g: B' \to B$ and the following diagrams commutes:

\[ \xymatrix{
& A \ox B' \ar[dl]_{1 \ox g} \ar[dr]^{f \ox 1} & \\
A \ox B \ar[dr]_{\psi_0} & & A' \ox B' \ar[ld]^{\psi'_0} \\
& D &}
& B' \ox A \ar[dl]_{g \ox 1} \ar[dr]^{1 \ox f} & \\
B \ox A \ar[dr]_{\psi_1} & & B' \ox A' \ar[ld]^{\psi'_1} \\
& D &}
\]

Compositon: $(f,g)(f',g') := (ff', g'g)$. Composition is well-defined as:

\[
\xymatrix{
&& A \ox B'' \ar[dl]_{1 \ox g'} \ar[dr]^{f \ox 1}   && \\
& A \ox B' \ar[dr]_{f' \ox 1}  \ar[dl]_{1 \ox g}  & & A' \ox B'' \ar[dl]^{1 \ox g'}  \ar[dr]^{f' \ox 1} \\
A \ox B \ar[drr]^{\psi_0} & & A' \ox B' \ar[d]^{\psi'_0} && A'' \ox B'' \ar[lld]_{\psi_0''} \\
& & D & &
\]

and similarly for the reverse Chu-maps $\psi_1$, $\psi'_1$ and $\psi_1''$.
The identity maps are $(1_A,1_B): (A, B, \psi_0,\psi_1) \to (A, B, \psi_0,\psi_1)$ as expected.

It is standard that ${\sf Chu}_\X(D)$  is a (non-commutative) $*$-autonomous category. Furthermore, it is cyclic because

$$~^{*}(A,B,\psi_0,\psi_1) = (A,B,\psi_0,\psi_1)^{*} = (B,A,\psi_1,\psi_0).$$

In addition, ${\sf Chu}_\X(D)$  is involutive with

$$\overline{(A,B,\psi_0,\psi_1)} := (\overline{A},\overline{B},\chi \overline{\psi_1}d,\chi \overline{\psi_0}d)$$

and $\overline{(f,g)} = (\overline{f},\overline{g})$.
Finally being involutive cyclic $*$-autonomous implies that one has a dagger!

In the case that $\X$ is a symmetric monoidal closed category we recapture the usual Chu construction [3], which we denote ${\sf Chus}_\X(D)$.  Consider the full subcategory
of Chu-objects with special Chu-cells of the form $(A,B, \psi,c_\otimes \psi)$ in which the symmetry map is used to obtain the second cell, this gives an inclusion
${\sf Chus}_\X(D) \to {\sf Chu}_\X(D)$.  We observe that when $\X$ is symmetric involutive that this subcategory is closed to the involution:

If $\X$  is an involutive symmetric monoidal closed category  and $d: \overline{D} \to D$ is an involutive object, then ${\sf Chus}_\X(D)$ is an involutive monoidal category.

It suffices to observe that the Chu-cells of $\overline{(A,B,\psi,c_\otimes\psi)}$ have the right form.  Using the coherence of the
involution with symmetry, the first Chu-cell of this object has  $\chi \overline{c_\otimes \psi}d = c_\ox \chi \overline{\psi}d$
which is exactly the symmetry map applied to the second Chu-cell of the object as desired.

Of course, to obtain a MUC we must first have an isomix category, in this regard we shall use the well-known fact that ${\sf Chus}_\X(I)$ is an isomix category simply because
the unit for tensor and par are the same (namely $\top = \bot = (I,I)$).  Of course, the tensor unit is always an involutive object since
$(\chi^{\!\!\!\circ})^{-1}: \overline{I} \to I$; therefore, this is immediately an involutive symmetric $*$-autonomous category.  Composing the involution with the dualizing functor gives us the dagger.   What remains is
to identify the objects which are in the core of ${\sf Chus}_\X(I)$ and to identify specific non-trivial preunitary objects.

A closed symmetric monoidal category is (degenerately) a compact linearly distributive category and thus there may be objects which have linear adjoints:
called nuclear objects.  They form a compact closed subcategory, thus in ${\sf Vec}_\mathbb{C}$ all the finite dimensional vector spaces are in the nucleus.  It is not hard then to
show that if $(\eta, \epsilon): A \dashv\!\!\!\dashv B$ that the object $(A,B,\epsilon,c_\otimes\epsilon)$ is in the core of ${\sf Chus}_\X(I)$.

We want to show that there are non-trivial examples of pre-unitary objects. To achieve this we consider an object $H$ for which $(e, n): H \dashv\!\!\!\dashv \overline{H}$
and such that $e$ satisfies:

\[ \xymatrix{\overline{\overline{H}} \otimes \overline{H} \ar[d]_{\varepsilon \otimes 1} \ar[r]^{\chi} & \overline{\overline{H} \otimes \overline{H}} \ar[dd]^{\overline{e}}\\
H \otimes \overline{H} \ar[d]_{e} \\
I \ar[r]_{\chi^{\!\!\!\circ}} & \overline{I} } \]

In ${\sf Vec}_{\mathbb{C}}$ $e: H \otimes \overline{H} \to \mathbb{C}$ is a “sesquilinear form” and the diagram above asserts that it is in addition a symmetric form.  Any finite dimensional
Hilbert space with its inner product satisfies the above conditions.

For such an object we note:

\begin{align*}
{ \overline{(H,\overline{H},e,c_\otimes e)}}^* & =  (\overline{H},\overline{\overline{H}},\chi\overline{c_\otimes e} (\chi^{\!\!\!\circ})^{-1},\chi\overline{e} (\chi^{\!\!\!\circ})^{-1})^* \\
& =  (\overline{\overline{H}},\overline{H},\chi\overline{e} (\chi^{\!\!\!\circ})^{-1},\chi\overline{c_\otimes e} (\chi^{\!\!\!\circ})^{-1})
\end{align*}

This makes

$$(\varepsilon^{-1},1) : (H,\overline{H},e,c_\otimes e) \to (\overline{\overline{H}},\overline{H},\chi\overline{e} (\chi^{\!\!\!\circ})^{-1},\chi\overline{c_\otimes e} (\chi^{\!\!\!\circ})^{-1})$$

a preunitay map.  Note that it is a Chu map by the commuting diagram above and as $\overline{\varepsilon} =\varepsilon$ we have

$$(\varepsilon^{-1},1) (1,\overline{\varepsilon}) = (\varepsilon^{-1},1) (1,\varepsilon) = (\varepsilon^{-1},\varepsilon)$$

where the latter is the involutor.

Of course, given such an object  $H$ one can take an object isomorphic to $\overline{H}$ to obtain a core object with a non-identity pre-unitary structure (which looks less familiar).  Thus, this rather long story ends up giving, by the forming the unitary core, examples which are neither compact nor having unitary structure which is
given (uniformly) by identity maps.

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