\[
E \underset{v}{\overset{u}{\rightleftarrows}} F,
\qquad i : \mathrm{id}_E \to vu, \quad j : uv \to \mathrm{id}_F,
\]

\[
\mathrm{Hom}_{E'}(X,Y) \longrightarrow \mathrm{Hom}_{F}(uX, uY)
\quad (\simeq \mathrm{Hom}(X, vuY))
\]

\[
\mathrm{Hom}_{F'}(P,Q) \longrightarrow \mathrm{Hom}(vP, vQ)
\quad (\simeq \mathrm{Hom}(uvP, Q))
\]

\[
\alpha(E') = \text{image essentielle de } u|E' = \overline{u(E')},
\qquad
\beta(F') = \text{image essentielle de } v|F' = \overline{v(F')}.
\]

\[
F_0 = \overline{u(E_0)}, \qquad E_0 = \overline{v(F_0)}
\]

\[
v(P) \xrightarrow{\ i(v(P))\ } vuv(P) \quad \text{(isomorphisme)}
\]

\[
\mathrm{Hom}_F(Q_0, P) \simeq \mathrm{Hom}_F(u_0 v_0 Q_0, P)
\quad \text{(car } u_0 v_0 \simeq \mathrm{id}\text{)}
\]

\[
\simeq \mathrm{Hom}_E(v_0 Q_0, vP) = \mathrm{Hom}_{E_0}(v_0 Q_0, v'P)
\simeq \mathrm{Hom}_{F_0}(u_0 v_0 Q_0, u_0 v' P)
\quad \text{($u_0$ pl. fid.)}
\]

\[
\simeq \mathrm{Hom}_{F_0}(Q_0, u_0 v' P)
\quad \text{(} u_0 v_0 \simeq \mathrm{id} \text{).]}
\]

\[
uvu(X) \xrightarrow{\ j(u(X))\ } u(X) \quad \text{(isomorphisme)},
\]

\[
\mathrm{Hom}_E(Y, X_0) \simeq \mathrm{Hom}_E(Y, v_0 u_0 X_0)
= \mathrm{Hom}_E(Y, v u_0 X_0)
\simeq \mathrm{Hom}_F(uY, u_0 X_0)
\]

\[
= \mathrm{Hom}_{F_0}(u'Y, u_0 X_0)
\simeq \mathrm{Hom}_{E_0}(v_0 u' Y, v_0 u_0 X_0)
= \mathrm{Hom}_{E_0}(v_0 u' Y, X_0). \qquad \text{cqfd}
\]

\[
E_0 \underset{v_0}{\overset{u_0}{\rightleftarrows}} F_0,
\]

\[
\varphi : E_1 \to (\mathrm{Ens}), \qquad
\psi : E_1^{\circ} \to (\mathrm{Ens}), \qquad \text{savoir}
\]

\[
\varphi(X) = \mathrm{Hom}(a, \beta X), \qquad
\psi(X) = \mathrm{Hom}(\beta X, a),
\]

\[
\varphi(X) \times \psi(Y) \longrightarrow \mathrm{Hom}(Y, X).
\]

\[
X \mapsto \mathrm{Hom}_F(\beta\alpha'(X), a) = \psi(\alpha'(X))
\]

\[
\mathrm{Hom}_E(X, \xi) \simeq \mathrm{Hom}_{E_1}(\alpha' X, \xi_1)
\quad (\simeq \mathrm{Hom}_E(X, \alpha\,\xi_1)),
\]

\[
(u : X \to Y) \in \Sigma
\iff \alpha'(X) \simeq \alpha'(Y)
\iff \mathrm{Hom}_{E_1}(\alpha'(Y), Z_1) \simeq
     \mathrm{Hom}_{E_1}(\alpha'(X), Z_1)
\ \ \text{pour } \forall\, Z_1 \in \mathrm{Ob}\,E_1
\]

\[
\iff \mathrm{Hom}_{E_1}(Y, \alpha Z_1) \simeq \mathrm{Hom}_{E_1}(X, \alpha Z_1)
\quad \text{pour } \forall\, Z_1 \in \mathrm{Ob}\,E_1.
\]

\[
(I, (L_i)_{i\in I}) \otimes (J, (M_j)_{j\in J})
= (I \sqcup J,\ (L_i;\ M_j)) \ \ldots
\]

\[
\Phi(A) \longrightarrow (\text{Ens finis, avec isom comme morphismes})
\]

\[
\Phi(A) = \coprod_n \Phi_n(A),
\qquad \Phi_0(A) = \{e\}, \quad \Phi_1(A) \simeq A.
\]

\[
\underline{\mathrm{Hom}}^{\otimes \mathrm{AUC}}(\Phi(A), C)
\longrightarrow \underline{\mathrm{Hom}}(A, C).
\]

\[
\pi : \Phi(C) \longrightarrow C, \qquad \pi = (\pi_n),
\]

\[
\Bigl(\coprod_{j\in J} I_j,\ (L_{ji})_{j\in J,\ i\in I_j}\Bigr)
\]

\[
\pi(\xi|J) \longrightarrow \mathbf{1}_C
\]

\[
\mathrm{inc},\ \mathrm{res} : \Phi^{c}(C) \longrightarrow \Phi(C)
\]

\[
\Pi \circ \mathrm{inc} \xrightarrow{\ c\ } \Pi \circ \mathrm{res}
\]

\[
L \otimes L' \simeq \mathbf{1},
\]

\[
L' \otimes L \simeq \mathbf{1},
\]

\[
\Pi(L, L') \simeq \mathbf{1}.
\]

\[
\varphi_{L,L'} \otimes \mathrm{id}_L \simeq \mathrm{id}_L \otimes \psi_{L',L}
\ :\ L \otimes L' \otimes L \xrightarrow{\ \sim\ } L,
\qquad \psi_{L',L} : L' \otimes L \simeq \mathbf{1}.
\]

\[
\varphi_{L,L'} \otimes \mathrm{id}_L\ :\ L \otimes L' \otimes L \to L,
\]

\[
\mathrm{id}_L \otimes \varphi_{L',L}
= \mathrm{id}_L \otimes (\varphi_{L,L'} \circ s_{L',L})
\ :\ L \otimes L' \otimes L \to L.
\]

\[
\underline{\mathrm{Hom}}(I; T, T') \longrightarrow
\underline{\mathrm{Hom}}(T, T')
\]

\[
\Big\downarrow \qquad \Big\downarrow
\]

\[
\Bigl[\ \underline{\mathrm{Hom}}_0(I; T, T') \longrightarrow
\underline{\mathrm{Hom}}_0(T, T')\ \Bigr] \quad (\text{resp. } \varprojlim)
\]

\[
T \longrightarrow \dot{\imath}.
\]

\[
\Gamma^{T} \times T \longrightarrow \dot{\imath}
\qquad (\text{resp. } \Gamma_0^{T} \times T \to \dot{\imath}_0).
\]

\[
u \mapsto u(\xi), \qquad \Gamma^{T} \longrightarrow C
\quad (\text{resp. } \Gamma_0^{T} \to C),
\]

\[
\mathrm{Hom}_{\mathrm{Ens}}(\rho(c), \varphi(c))
= \mathrm{Hom}_{\mathrm{Ens}}(x, \emptyset) = \emptyset\ !
\]

\[
\underline{\mathrm{Hom}}(C, F)
\underset{\beta}{\overset{\alpha}{\rightleftarrows}}
\underline{\mathrm{Hom}}(E, F)
\]

\[
\alpha(\varphi)(X) = \varinjlim_{Y \in C/X} \varphi(Y),
\qquad
\beta(\psi)(Y) = \psi(Y), \quad \text{i.e. } \beta(\psi) = \psi | C
= \psi \circ i.
\]

\[
\beta\alpha(\varphi)(Y) = \alpha(\varphi)(Y)
= \varinjlim_{Z \in C/Y} \varphi(Z) = \varphi(Y),
\quad \text{i.e. } \beta\alpha(\varphi) \simeq \varphi, \quad
\beta\alpha \simeq \mathrm{id}_{\underline{\mathrm{Hom}}(C,F)}.
\]

\[
\alpha\beta(\psi)(X) = \varinjlim_{Y \in C/X} \beta\psi(Y)
= \varinjlim_{Y \in C/X} \psi(Y) \longrightarrow \psi(X),
\quad \text{i.e. } \alpha\beta \to \mathrm{id}_{\underline{\mathrm{Hom}}(E,F)}.
\]

\[
\mathrm{Hom}(\alpha\varphi, \psi) \simeq \mathrm{Hom}(\varphi, \beta\psi).
\]

\[
\varphi(Y) \xleftarrow{\ \sim\ } \varinjlim \varphi(Y_i).
\]

\[
\beta' : \underline{\mathrm{Hom}}'(E, F) \longrightarrow
\underline{\mathrm{Hom}}'(C, F).
\]

\[
\alpha\beta(\psi)(X) = \varinjlim_{Y \in C/X} \psi(Y)
\xrightarrow{\ \sim\ } \psi(X)
\quad \text{car } X = \varinjlim_{Y \in C/X} Y,\ C
\]

\[
\beta' : \underline{\mathrm{Hom}}'(E, F) \longrightarrow
\underline{\mathrm{Hom}}'(C, F)
\]

\[
\varinjlim_i \alpha\varphi(Y_i)
= \varinjlim_i \bigl(\varinjlim_j \varphi(Y_{ij})\bigr)
= \varinjlim_{ij} \varphi(Y_{ij}),
\]

\[
\varinjlim \varphi(Y_i) \xrightarrow{\ \sim\ } \beta\varphi(Y).
\]