\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\underline{\mathrm{Hom}}(B^{\circ}, \widehat{A})
  & (A\times B)^{\wedge} \arrow[l, "\approx"'] \arrow[r, "\approx"] \arrow[d, "\approx"]
  & \underline{\mathrm{Hom}}(A^{\circ}, \widehat{B}) \arrow[d, "\approx"] \\
  & \underline{\mathrm{Hom}}^{!!}\bigl(\widehat{A}^{\circ} \times \widehat{B}^{\circ},\ (\mathrm{Ens})\bigr) = \mathrm{Bifais}(\widehat{A}, \widehat{B})
  & \underline{\mathrm{Hom}}^{!}(\widehat{A}^{\circ}, \widehat{B}) \\
  & \underline{\mathrm{Hom}}_{!}(A^{\vee}, \widehat{B}) \arrow[u, "\approx"']
  &
\end{tikzcd}

\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
(\widehat{B}_{\to}) & & & & & & (\widehat{A}_{\to}) \\
& \underline{\mathrm{Hom}}^{!}(\widehat{B}^{\circ},\widehat{A}) \arrow[rr, bend left=22] \arrow[dd, bend right=22] & & \underline{\mathrm{Hom}}^{!!}(\widehat{A}^{\circ},\widehat{B}^{\circ};\,\mathrm{Ens}) & & \underline{\mathrm{Hom}}^{!}(\widehat{A}^{\circ},\widehat{B}) \arrow[ll, bend right=22] \arrow[dd, bend left=22] \arrow[dl] & \\
& & \underline{\mathrm{Hom}}(B^{\circ},\widehat{A}) \arrow[ul] & & \underline{\mathrm{Hom}}(A^{\circ},\widehat{B}) & & \\
& \underline{\mathrm{Hom}}_{!}^{!}(A^{\vee},\widehat{B}^{\circ};\,\mathrm{Ens}) & & \substack{(A\times B)^{\wedge} \\ =\, \underline{\mathrm{Hom}}(A^{\circ},B^{\circ};\,\mathrm{Ens})} \arrow[ul] \arrow[ur] \arrow[dl] \arrow[dr] \arrow[uu, no head, "\wedge\wedge" description] \arrow[ll, no head, "\vee\wedge" description] \arrow[rr, no head, "\wedge\vee" description] \arrow[dd, no head, "\vee\vee" description] & & \underline{\mathrm{Hom}}^{!}_{!}(\widehat{A}^{\circ},B^{\vee};\,\mathrm{Ens}) & \\
& & \underline{\mathrm{Hom}}(A^{\circ},\widehat{B}) \arrow[dl] & & \underline{\mathrm{Hom}}(B^{\circ},\widehat{A}) \arrow[dr] & & \\
& \underline{\mathrm{Hom}}_{!}(A^{\vee},\widehat{B}) \arrow[uu, bend left=22] \arrow[rr, bend right=22] & & \underline{\mathrm{Hom}}_{!!}(A^{\vee},B^{\vee};\,\mathrm{Ens}) & & \underline{\mathrm{Hom}}_{!}(B^{\vee},\widehat{A}) \arrow[uu, bend right=22] \arrow[ll, bend left=22] & \\
(A^{\vee}_{\to}) & & & & & & (B^{\vee}_{\to})
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
A^{\vee\circ} \arrow[rr, "\Psi_{A}"] & & \widehat{A} \\
& A \arrow[ul, hook] \arrow[ur, hook'] &
\end{tikzcd}

\begin{tikzcd}[column sep=normal, row sep=normal]
\widehat{A} \arrow[r, hook] \arrow[d, "f_{!}"']
  & \widetilde{A} \arrow[d, Rightarrow, "\widetilde{f}"]
  & A^{\vee\circ} \arrow[l, hook'] \arrow[d, "f^{\circ*\circ}"] \\
\widehat{B} \arrow[r, hook]
  & \widetilde{B}
  & B^{\vee\circ} \arrow[l, hook']
\end{tikzcd}