\[
H_{A}\bigl(f^{*}(\Psi_{\varepsilon}),\ \Phi^{\varepsilon}\bigr)
\;\simeq\;
H_{B}\bigl(\Psi_{\varepsilon},\ f^{\varepsilon}_{!}(\Phi^{\varepsilon})\bigr)
\]

\[
H_{A}\bigl(f^{*}(b),\ a\bigr) \;\simeq\; H_{B}\bigl(b,\ f^{\varepsilon}(a)\bigr)
= \mathrm{Hom}_{B}\bigl(b,\ f(a)\bigr),
\qquad
f^{\varepsilon}(a) = \varinjlim \cdots
\]

\[
\underline{\mathrm{Hom}}_{A}(X,Y) \overset{\mathrm{def}}{=}
   p_{B*}\,\underline{\mathrm{Hom}}_{A\times B}(X,Y),
\qquad
\underline{\mathrm{Hom}}_{B}(X,Y) \overset{\mathrm{def}}{=}
   p_{A*}\,\underline{\mathrm{Hom}}_{A\times B}(X,Y)
\]

\[
\mathrm{Hom}_{A\times B}(X,Y)
\simeq \Gamma_{A\times B}\,\underline{\mathrm{Hom}}_{A\times B}(X,Y)
\simeq \Gamma_{A}\,\underline{\mathrm{Hom}}_{A}(X,Y)
\simeq \Gamma_{B}\,\underline{\mathrm{Hom}}_{B}(X,Y),
\qquad X, Y \in (A\times B)^{\wedge}
\]

\[
{}^{s}X(L) = \underline{\mathrm{Hom}}_{B}\bigl(p_{B}^{*}L,\ X\bigr)
           = p_{A*}\,\underline{\mathrm{Hom}}\bigl(p_{B}^{*}L,\ X\bigr),
\qquad
{}^{d}X(M) = \underline{\mathrm{Hom}}_{A}\bigl(p_{A}^{*}M,\ X\bigr)
           = p_{B*}\,\underline{\mathrm{Hom}}\bigl(p_{A}^{*}M,\ X\bigr)
\]

\[
X(L,M) = \mathrm{Hom}\bigl(L \boxtimes M,\ X\bigr)
\simeq \mathrm{Hom}\bigl(L,\ {}^{d}X(M)\bigr)
\simeq \mathrm{Hom}\bigl(M,\ {}^{s}X(L)\bigr),
\qquad\text{où } L \boxtimes M = p_{B}^{*}L \times p_{A}^{*}M .
\]

\[
\underline{\mathrm{Hom}}_{A\times B}\bigl(L\boxtimes M',\ L'\boxtimes N'\bigr)
\simeq \underline{\mathrm{Hom}}_{A}(L,L') \boxtimes
       \underline{\mathrm{Hom}}_{B}(M',N')
\qquad \text{p.ex.}
\]

\[
\underline{\mathrm{Hom}}_{A\times B}\bigl(L\boxtimes M',\ L'\boxtimes e_{B}\bigr)
\simeq \underline{\mathrm{Hom}}_{A}(L,L') \boxtimes e_{B}
= p_{B}^{*}\,\underline{\mathrm{Hom}}_{A}(L,L')
\]

\[
I = \coprod_{j\in J} I_{j}, \qquad J = J^{+} \amalg J^{-},
\qquad I^{+} = \coprod_{j\in J^{+}} I_{j}, \quad
I^{-} = \coprod_{j\in J^{-}} I_{j}
\]

\[
\widehat{A_{I}} \;\simeq\;
\underline{\mathrm{Hom}}^{!!}\Bigl(\prod_{j\in J^{-}} A_{I_{j}}^{\vee}\,,\
\prod_{j'\in J^{+}} \widehat{A_{I_{j'}}}^{\circ}\,;\ \mathrm{Ens}\Bigr)
\]

\[
\underline{\mathrm{Hom}}_{!}\Bigl(\prod_{j\in J^{-}} A_{I_{j}}^{\vee},\
\widehat{A_{I^{+}}}\Bigr),
\qquad
\underline{\mathrm{Hom}}^{!}\Bigl(\prod_{j'\in J^{+}}
\widehat{A_{I_{j'}}}^{\circ},\ A_{I^{-}}^{\vee}\Bigr)
\]

\[
\boxed{\ H_{A}(\Phi,\Psi)
= \mathrm{Hom}_{\widehat{A}}\bigl(\Phi,\ \Psi_{A}(\Psi)\bigr)
\simeq \mathrm{Hom}_{A^{\vee}}\bigl(\Psi,\ \varphi_{A}(\Phi)\bigr)\ }
\]

\[
\Phi \longrightarrow \Psi_{A}\,\varphi_{A}^{\circ}(\Phi)
\ \text{dans } \widehat{A},
\qquad
\Psi \longrightarrow \varphi_{A}\,\Psi_{A}^{\circ}(\Psi)
\ \text{dans } A^{\vee} ;
\]

\[
\boxed{\ H_{A}(a,b) = \mathrm{Hom}_{A}(a,b)\ }
\]

\[
H_{A}(\Phi, b) \simeq \mathrm{Hom}_{\widehat{A}}(\Phi,\ b),
\qquad
H_{A}(a, \Psi) \simeq \mathrm{Hom}_{A^{\vee}}(\Psi,\ a)
= \mathrm{Hom}_{A^{\vee\circ}}(a,\ \Psi)
\]

\[
\boxed{\ H^{A} :\ (\Phi,\Psi) \longmapsto \Phi \star_{A} \Psi\ ;\quad
\widehat{A} \times A^{\vee} \longrightarrow \mathrm{Ens}\ }
\]

\[
\boxed{\ H^{A}(a,b) = \mathrm{Hom}_{A}(b,a)\ }
\]

\[
\boxed{\ \widehat{A} \xrightarrow{\ \sim\ }
\underline{\mathrm{Hom}}_{!}(A^{\vee}, \mathrm{Ens}),
\qquad
A^{\vee} \xrightarrow{\ \sim\ }
\underline{\mathrm{Hom}}_{!}(\widehat{A}, \mathrm{Ens})\ }
\]

\[
\Phi \star a = \Phi(a), \qquad a \star \Psi = \Psi(a)
\]

\[
H^{A}(\Phi,\Psi) = \varphi^{A}(\Phi)(\Psi) \simeq \Psi^{A}(\Psi)(\Phi)
\]

\[
\Bigl(\prod_{i\in I} A_{i}\Bigr)^{\wedge},
\qquad
A_{1}^{\circ} \times A_{2}^{\circ} \times \cdots \times A_{n}^{\circ}
\to \mathrm{Ens},
\qquad
\widehat{A_{1}}^{\circ} \times \widehat{A_{2}} \times A_{3}^{\vee}
\times \cdots
\]

\[
X_{1} \otimes X_{2} \simeq \mathrm{Hom}(X_{1}^{\vee}, X_{2})
\simeq \mathrm{Hom}(X_{2}^{\vee}, X_{1})
\]

\[
F(X_{1}\boxtimes X_{2})
\simeq \underline{\mathrm{Hom}}^{!!}\bigl(F(X_{1})^{\circ},\ F(X_{2})^{\circ};\
\mathrm{Ens}\bigr)
\simeq \cdots
\simeq \mathrm{Hom}\bigl(F(X_{1})^{\vee},\ F(X_{2})\bigr)
\simeq \mathrm{Hom}\bigl(F(X_{2})^{\vee},\ F(X_{1})\bigr)
\]

\[
F(X_{I})
\simeq \underline{\mathrm{Hom}}^{(1)}\Bigl(\prod_{j\in J} F(X_{I_{j}})^{\circ},\
\mathrm{Ens}\Bigr)
\simeq \underline{\mathrm{Hom}}_{(1)}\Bigl(\prod_{j\in J}
F(X_{I_{j}}^{\vee}),\ \mathrm{Ens}\Bigr)
\simeq \cdots
\]

\[
\cdots \simeq \underline{\mathrm{Hom}}_{!}\Bigl(\prod_{j\in J^{-}}
F(X_{I_{j}}),\ \prod_{j'\in J^{+}} F(X_{I_{j'}})\Bigr)
\simeq \underline{\mathrm{Hom}}^{!}\Bigl(\prod_{j'\in J^{+}}
F(X_{I_{j'}})^{\circ},\ \prod_{j\in J^{-}} F(X_{I_{j}}^{\vee})\Bigr)
\]

\[
\bigl\langle \Phi_{*},\ f^{\circ*}(\Psi^{*}) \bigr\rangle_{A}
\simeq \bigl\langle f_{!}(\Phi_{*}),\ \Psi^{*} \bigr\rangle_{B},
\qquad
\bigl\langle f^{*}(\Psi_{*}),\ \Phi^{*} \bigr\rangle_{A}
\simeq \bigl\langle \Psi_{*},\ f^{\circ}_{!}(\Phi^{*}) \bigr\rangle_{B}
\]

\[
H_{x} \in \widehat{A}, \quad H_{x}(a) = \mathrm{Hom}_{B}(a, x) ;
\qquad
H^{x} \in A^{\vee}, \quad H^{x}(a) = \mathrm{Hom}_{B}(x, a)
\]

\[
H_{x}(a) \times H^{x}(b) \xrightarrow{\ \alpha_{x}\ } \mathrm{Hom}_{A}(a, b)
\]

\[
\alpha_{a,b}\bigl(u,\ \varphi^{*}(u')\bigr)
= \alpha'_{a,b}\bigl(\varphi_{*}(u),\ u'\bigr)
\]

\[
\widehat{A} \xrightarrow{\ \approx\ }
\underline{\mathrm{Hom}}_{!}(A^{\vee}, \mathrm{Ens}),
\qquad
A^{\vee\circ} \xrightarrow{\ \approx\ }
\underline{\mathrm{Hom}}_{\mathrm{rep}}(\widehat{A}, (\mathrm{Ens}))
\]

\[
\bigl(\mathrm{Kar}(A^{\circ})\bigr)^{\circ} \;\simeq\; \mathrm{Kar}(A)
\]

\[
A = \widehat{A} \cap A^{\vee\circ}
\]

\[
\underline{\mathrm{Hom}}(A, M)
\simeq \underline{\mathrm{Hom}}_{!}(\widehat{A}, M)
\]

\[
\underline{\mathrm{Hom}}(A, M) \simeq
\underline{\mathrm{Hom}}^{!}(\widehat{A}, M)
\]

\[
\underline{\mathrm{Hom}}(B^{\circ}, M) \simeq
\underline{\mathrm{Hom}}^{!}(\widehat{B}^{\circ}, M)
\]

\[
\mathcal{F}(X, M)^{\circ} \simeq \check{\mathcal{F}}(X, M^{\circ}),
\qquad
\bigl(\underline{\mathrm{Hom}}^{!}(A, M)\bigr)^{\circ}
\simeq \underline{\mathrm{Hom}}_{!}(A^{\circ}, M^{\circ})
\]

\[
\mathcal{F}(X, M) \approx \check{\mathcal{F}}(X^{\circ}, M),
\qquad
\underline{\mathrm{Hom}}^{!}(A^{\circ}, M)
\simeq \underline{\mathrm{Hom}}_{!}(A, M^{\circ})^{\circ}
\]