\[
X_*, Y_* \in \mathrm{Ob}\, \underbrace{(A \times \Delta)^{\wedge}}
\simeq \underline{\mathrm{Hom}}(\Delta^{\circ}, A^{\wedge})
\simeq \underline{\mathrm{Hom}}(A^{\circ}, \Delta^{\wedge})
\]

\[
\begin{cases}
\mathrm{Hom}_{A \times \Delta}(X_*, Y_*) = \Gamma_{A \times \Delta}\bigl(\underline{\mathrm{Hom}}_{A \times \Delta}(X_*, Y_*)\bigr)\\
\underline{\mathrm{Hom}}_A(X_*, Y_*) = p_{\Delta*}\, \underline{\mathrm{Hom}}_{A \times \Delta}(X_*; Y_*)\\
\underline{\mathrm{Hom}}_{\Delta}(X_*, Y_*) = p_{A*}\, \underline{\mathrm{Hom}}_{A \times \Delta}(X_*, Y_*)
\end{cases}
\]

\[
\begin{cases}
p_{A*} X_* = (\Gamma_A X_n)_n\\
p_{\Delta*} X_* = X_0
\end{cases}
\qquad
(q_{\Delta} p_A)_*(X_*) = (q_A p_{\Delta})_* X_* = \Gamma_A X_0
\]

\[
\begin{aligned}
&\text{I}\quad \bigl(\underline{\mathrm{Hom}}_A(L, X_n)\bigr)_{n \in \mathbb{N}}
= \underline{\mathrm{Hom}}_{A \times \Delta}\bigl(p_{\Delta}^{*}(L), X_*\bigr)
&&\text{hence, by applying } p_{A*}\\
&\text{II}\quad \bigl(\mathrm{Hom}_A(L, X_n)\bigr)_{n \in \mathbb{N}}
= \underline{\mathrm{Hom}}_{\Delta}\bigl(p_{\Delta}^{*}(L), X_*\bigr)
\end{aligned}
\]

\[
\begin{aligned}
\underline{\mathrm{Hom}}_B(X, Y) &\overset{\text{déf}}{=} p_{A*}\, \underline{\mathrm{Hom}}_{A \times B}(X, Y)\\
\underline{\mathrm{Hom}}_A(X, Y) &\overset{\text{déf}}{=} p_{B*}\, \underline{\mathrm{Hom}}_{A \times B}(X, Y)
\end{aligned}
\]

\[
\mathrm{Hom}(X, Y) \simeq \Gamma_A\, \underline{\mathrm{Hom}}_A(X, Y)
\simeq \Gamma_B\, \underline{\mathrm{Hom}}_B(X, Y)
\]

\[
\begin{array}{ccc}
\Gamma''_b : (A \times B)^{\wedge} & \longrightarrow & A^{\wedge}\\
\downarrow & & \|\\
\simeq \underline{\mathrm{Hom}}(B^{\circ}, A^{\wedge}) & \longrightarrow & A^{\wedge}\\
X'' & \longmapsto & X''(b)
\end{array}
\qquad \text{déduit par } (A \times B)^{\wedge} \simeq
\underline{\mathrm{Hom}}(A^{\circ}, B)
\]

\[
\Gamma''_b(X) \simeq p_{B*}\, \underline{\mathrm{Hom}}\bigl(p_A^{*}(b), X\bigr)
\qquad \text{fonctoriel en } b, X
\]

\[
\bigl(\simeq \underline{\mathrm{Hom}}_A(p_A^{*}(b), X)\bigr),
\]

\[
\begin{aligned}
\underset{X(a, b)}{\underset{\|}{\mathrm{Hom}_{A^{\wedge}}\bigl(a, \Gamma''_b(X)\bigr)}}
&\simeq \mathrm{Hom}_{A^{\wedge}}\bigl(a, p_{B*}\, \underline{\mathrm{Hom}}(p_A^{*}(b), X)\bigr)
&&\text{fonctoriel en } a, X, b\\
&\simeq \mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_B^{*}(a), \underline{\mathrm{Hom}}(p_A^{*}(b), X)\bigr)\\
&\simeq \mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_B^{*}(a) \times p_A^{*}(b), X\bigr)
&&\text{OK}
\end{aligned}
\]

\[
\begin{array}{ccc}
\Gamma'_a : (A \times B)^{\wedge} & \longrightarrow & B^{\wedge}\\
\wr & &\\
\underline{\mathrm{Hom}}(A^{\circ}, B^{\wedge}) & &\\
X' & \longmapsto & X'(a)
\end{array}
\]

\[
\Gamma'_a(X) \simeq p_{A*}\bigl(\underline{\mathrm{Hom}}(p_B^{*}(a), X)\bigr)
\]

\[
\varepsilon_b : B^{\wedge} \to \mathrm{Ens}, \qquad
\varepsilon_b(G) = \mathrm{Hom}(b, G) = G(b)
\]

\[
\varepsilon_M : B^{\wedge} \to \mathrm{Ens}, \qquad
G \mapsto \varepsilon_M(G) = \mathrm{Hom}(M, G)
\]

\[
\Gamma''_M : \underset{\underline{\mathrm{Hom}}(A^{\circ}, B^{\wedge})}{\underset{\cap}{X''}}
\longmapsto \varepsilon_M \circ \Gamma''_M
\]

\[
\Gamma''_M(X) \simeq p_{B*}\, \underline{\mathrm{Hom}}\bigl(p_A^{*}(M),
X\bigr) \qquad \text{fonctoriel en } M, X
\]

\[
\begin{array}{ccc}
\mathrm{Hom}_{A^{\wedge}}\bigl(a, \Gamma''_M(X)\bigr) & \simeq &
\mathrm{Hom}_{A^{\wedge}}\bigl(a, p_{B*}\, \underline{\mathrm{Hom}}(p_A^{*}(M), X)\bigr)\\
\| & & \wr\\
\Gamma''_M(X)(a) & &
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_B^{*}(a), \underline{\mathrm{Hom}}(p_A^{*}(M), X)\bigr)\\
\| & & \wr\\
\mathrm{Hom}_{B^{\wedge}}\bigl(M, X''(a)\bigr) & &
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_B^{*}(a) \times p_A^{*}(M), X\bigr)\\
\wr & &\\
p_{A*}\bigl(\underline{\mathrm{Hom}}(p_B^{*}(a), X)\bigr) & &\\
\| & &\\
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_A^{*}(M), \underline{\mathrm{Hom}}(p_B^{*}(a), X)\bigr) & &\\
\| & &\\
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_A^{*}(M) \times p_B^{*}(a), X\bigr) & &
\end{array}
\]

\[
\underline{\mathrm{Hom}}(p_A^{*}(M), X) \overset{\text{déf}}{=} X^{M}
\qquad \text{pour } \forall\, M \in \mathrm{Ob}\, B^{\wedge}
\]

\[
\begin{aligned}
\underline{\mathrm{Hom}}_B(X, Y^{M})
&\simeq p_{A*}\, \underbrace{\underline{\mathrm{Hom}}\bigl(X, \underline{\mathrm{Hom}}(p_A^{*}(M), Y)\bigr)}\\
&\simeq p_{A*}\bigl(\underline{\mathrm{Hom}}(X \times p_A^{*}(M), Y)\bigr)\\
&\simeq \underline{\mathrm{Hom}}_B(X \times p_A^{*}(M), Y)
\end{aligned}
\]

\[
\underline{\mathrm{Hom}}_B(X, Y^{M}) \simeq
\underline{\mathrm{Hom}}_B(\underbrace{X \times p_A^{*}(M)}, Y)
\]

\[
b \longmapsto \underbrace{X(b) \times M(b)} = \Bigl\{ a \longmapsto
\underbrace{\mathrm{Hom}_A\bigl(a, X(b)\bigr)} \times M(b) \Bigr\}
\]

\[
X(b)(a) = X(a, b)
\]

\[
X \otimes M \in \underline{\mathrm{Hom}}(B^{\circ}, A) \quad \text{par}
\]

\[
(X \otimes M)(b) = \coprod^{A}_{\sigma \in M(b)} X(b)
\quad \Bigl( \xrightarrow[\;i_b\;]{\;\text{nat}\;}
\coprod^{A^{\wedge}}_{\sigma \in M(b)} X(b) = \bigl(X \times p_A^{*}(M)\bigr)(b) \Bigr)
\]

\[
\underline{\mathrm{Hom}}_B(X, Y^{M}) \simeq \underline{\mathrm{Hom}}_B(X
\otimes M, Y) \qquad \text{fonctoriel}
\]

\[
\begin{aligned}
X(a, b) &= \Gamma_{A \times B}\,
\underline{\mathrm{Hom}}\bigl(p_B^{*}(a) \times p_A^{*}(b), X\bigr)\\
&= \Gamma_{A \times B}\bigl(\underline{\mathrm{Hom}}_{A \times B}(a \boxtimes b, X)\bigr)
\end{aligned}
\qquad
\begin{cases}
= \Gamma_A\, p_{B*}\, \underline{\mathrm{Hom}} \\
= \Gamma_B\, p_{A*}\, \underline{\mathrm{Hom}}
\end{cases}
\]

\[
{}^{g}X(a) = p_{B*}\, \underline{\mathrm{Hom}}_{A \times B}({}^{g}a, X)
\qquad
{}^{d}X(b) = p_{B*}\, \underline{\mathrm{Hom}}_{A \times B}({}^{d}b, X)
\]

\[
\begin{aligned}
X(L, M) &= \underline{\mathrm{Hom}}_{A \times B}(L \boxtimes M, X)
 = \Gamma_{A \times B}\, \underline{\mathrm{Hom}}(L \boxtimes M, X)\\
{}^{g}X(L) &= p_{A*}\, \underline{\mathrm{Hom}}_{A \times B}({}^{g}L, X)
 = \underline{\mathrm{Hom}}_B({}^{g}L, X)\\
{}^{d}X(M) &= \underbrace{p_{B*}\, \underline{\mathrm{Hom}}_{A \times B}({}^{d}M, X)}_{X^{M}}
 = \underline{\mathrm{Hom}}_A({}^{d}M, X)
\end{aligned}
\]

\[
\underline{\mathrm{Hom}}_B(X, Y^{M}) \simeq \underline{\mathrm{Hom}}_B(X \otimes M, Y)
\]

\[
\begin{aligned}
(A \times B)^{\wedge} &\overset{\text{déf}}{=}
\underline{\mathrm{Hom}}\bigl((A \times B)^{\circ}, (\mathrm{Ens})\bigr)
\simeq \underline{\mathrm{Hom}}\bigl(A^{\circ} \times B^{\circ}, (\mathrm{Ens})\bigr)\\
&\simeq \underline{\mathrm{Hom}}(A^{\circ}, B^{\wedge})
\simeq \underline{\mathrm{Hom}}_{!}(A^{\wedge\circ}, B^{\wedge})\\
&\simeq \underline{\mathrm{Hom}}(B^{\circ}, A^{\wedge})
\simeq \underline{\mathrm{Hom}}_{!}(B^{\wedge\circ}, A^{\wedge})\\
&\simeq \underline{\mathrm{Hom}}_{!!}\bigl((A^{\wedge})^{\circ} \times (B^{\wedge})^{\circ}, \mathrm{Ens}\bigr)
\simeq \underline{\mathrm{Hom}}_{!}\bigl((A^{\wedge} \times B^{\wedge})^{\circ}, \mathrm{Ens}\bigr)
\end{aligned}
\]

\[
(L, M) \longmapsto X(L, M) \quad :
\underset{(A^{\wedge} \times B^{\wedge})^{\circ}}{\underbrace{A^{\wedge\circ} \times B^{\wedge\circ}}}
\longrightarrow (\mathrm{Ens})
\]

\[
X(L, M) = \mathrm{Hom}(L \boxtimes M, X) \quad \text{où} \quad
L \boxtimes M = p_B^{*}(L) \times p_A^{*}(M) .
\]

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

\[
\overset{\text{déf}}{\|} \qquad
\mathrm{Hom}(L' \boxtimes M', L \boxtimes M)
\]

\[
\begin{array}{c}
\mathrm{Hom}_{A \times B}\bigl(\underbrace{(a \boxtimes b) \times (L' \boxtimes M')}_{(a \times L') \boxtimes (b \times M')}, L \boxtimes M\bigr)\\
\|\\
\mathrm{Hom}_A(a \times L', L) \times \mathrm{Hom}_B(b \times M', M)
\end{array}
\]

\[
\begin{array}{c}
\mathrm{Hom}_A\bigl(a, \underline{\mathrm{Hom}}(L', L)\bigr) \times \mathrm{Hom}_B\bigl(b, \underline{\mathrm{Hom}}(M', M)\bigr)\\
\|\\
\mathrm{Hom}_A(a \times L', L) \times \mathrm{Hom}_B(b \times M', M)
\end{array}
\]

\[
\begin{aligned}
(X^{\Delta_n})_0 &= p_{B*}\bigl(\underline{\mathrm{Hom}}(p_A^{*}(\Delta_n), X)\bigr) = X_n\\
(X^{\dot{\Delta}_n})_0 &= p_{B*}\, \underline{\mathrm{Hom}}(\dot{\Delta}_n, X) = \varprojlim\,(X_{n-1}, X_{n-2}\ \ldots)
\end{aligned}
\]

\[
X^{\Delta_n} \longrightarrow X^{\dot{\Delta}_n} \underset{Y^{\dot{\Delta}_n}}{\times} Y^{\Delta_n}
\qquad\qquad
X_n \longrightarrow (X^{\dot{\Delta}_n})_0 \underset{(Y^{\dot{\Delta}_n})_0}{\times} Y_n
\]

\[
\underline{\mathrm{Hom}}(\dot{\Delta}_n, a)
\qquad
\underline{\mathrm{Hom}}(M, a) \overset{??}{\Longrightarrow} a
\]

\[
\mathrm{Hom}(X \times M, a) \simeq \mathrm{Hom}(X, -)
\]

\[
\mathrm{Hom}\bigl(\alpha \boxtimes (\beta \times M), \alpha \boxtimes e_B\bigr)
\simeq \mathrm{Hom}(\alpha \boxtimes \beta, \alpha \boxtimes e_B)
\]

\[
\underline{\mathrm{Hom}}(L' \boxtimes M', L \boxtimes e_{B^{\wedge}})
\simeq \underline{\mathrm{Hom}}(L', L) \boxtimes e_{B^{\wedge}}
\]

\[
\underline{\mathrm{Hom}}(e_{A^{\wedge}} \boxtimes M, L \boxtimes e_{B^{\wedge}})
\simeq L \boxtimes e_{B^{\wedge}}
\]

\[
\underline{\mathrm{Hom}}(P, Q) \longleftarrow Q
\]

\[
L^{M} = L \qquad \text{pour } L \in A^{\wedge},\ M \in \mathrm{Ob}\, B^{\wedge}
\]

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

\[
\begin{aligned}
\underline{\mathrm{Hom}}(A^{\circ}, M) &\simeq \underline{\mathrm{Hom}}_{!}(A^{\circ\wedge}, M)
\simeq \Bigl(\underline{\mathrm{Hom}}^{!}\bigl(\underbrace{A^{\circ\wedge\circ} = A^{\vee\circ}}, M^{\circ}\bigr)\Bigr)^{\circ}\\
\wr\quad &\\
\underline{\mathrm{Hom}}(A, M^{\circ})^{\circ} &\simeq \bigl(\underline{\mathrm{Hom}}_{!}(A^{\wedge}, M^{\circ})\bigr)^{\circ}
= \underline{\mathrm{Hom}}^{!}(A^{\wedge\circ}, M)
\end{aligned}
\]

\[
\underline{\mathrm{Hom}}_{!}(A^{\vee}, M)^{\circ} \simeq \underline{\mathrm{Hom}}_{!}(A^{1}, M^{\circ})
\]

\[
\underline{\mathrm{Hom}}(A^{\circ}, M) \simeq \underline{\mathrm{Hom}}_{!}(A^{\circ\wedge}, M)
= \underline{\mathrm{Hom}}_{!}(A^{\vee}, M)
\]

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

\[
(\mathrm{Cat}) \longrightarrow \mathrm{Hot}(\underline{W}), \qquad
A^{\wedge} \longrightarrow \mathrm{Hot}_A \simeq \mathrm{Hot}(\underline{W})
\quad (A \text{ a test category})
\]

\[
X_0 \xrightarrow{\;f_0\;} X_1 \xrightarrow{\;f_1\;} X_2 \cdots
\]

\[
u_\infty : X'_\infty \longrightarrow X_\infty .
\]

\[
(*) \qquad \mathrm{Hom}_{\mathrm{Hot}}(X_\infty, Y) \xrightarrow{\;\sim\;}
\varprojlim_i \mathrm{Hom}_{\mathrm{Hot}}(X_i, Y) ,
\]

\[
\mathrm{Hom}_{\mathrm{Hot}}(X, Y) \simeq \overline{\mathrm{Hom}}(X, Y)
\]

\[
\varphi i \overset{k}{\sim} g \qquad p\varphi \overset{h}{\sim} g'
\]

\[
\overline{\varphi} \sim \varphi , \qquad \varphi i = g, \quad p\varphi = g' \ ?
\]

\[
k : g_0 = \varphi i \longrightarrow g_1 = g
\]

\[
h : g'_0 = p\varphi \longrightarrow g'_1 = g' .
\]

\[
h \circ (i \otimes J) = v .
\]

\[
\operatorname{Hom}(X_\infty, Y) \longrightarrow
\varprojlim \operatorname{Hom}(X_i, Y)
\]

\[
f, f' : X_\infty \longrightarrow Y, \qquad
f_i = f \,|\, X_i, \quad f'_i = f' \,|\, X_i ;
\]

\[
\underbrace{X_\infty \times \Delta(1)}_{B}
\ \underset{\text{cofib.}}{\supset}\
\underbrace{X_\infty \sqcup X_\infty}_{A} ,
\]

\[
\begin{array}{ccc}
B & \supset & A \\
\vdots & & \vdots \\
B_{i+1} & \supset & A_{i+1} \\
\text{cofib}\ \cup & & \cup\ \text{cofib} \\
B_i & \underset{\text{cofib.}}{\supset} & A_i
\end{array}
\]

\[
B'_{i+1} \overset{\text{def}}{=} B_i \sqcup_{A_i} A_{i+1}
\]

\[
\begin{array}{c}
\Delta(1) \times X_{i+1} \\
\cup \\
\bigl(\Delta(1) \times X_i\bigr) \cup \bigl(\{0\} \times X_{i+1}\bigr) \cup \bigl(\{1\} \times X_{i+1}\bigr)
\end{array}
\]

\[
Y \Leftarrow Z_0 \hookrightarrow Z_1 \hookrightarrow Z_2 \cdots
\]