Proposition 3.3.5.6. Let $X$ and $Y$ be simplicial sets, where $Y$ is a Kan complex. Then the bijection
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sd}(X), Y) \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{Ex}(Y) ) \]
respects homotopy. That is, for every pair of maps $f,g: \operatorname{Sd}(X) \rightarrow Y$ having counterparts $f',g': X \rightarrow \operatorname{Ex}(Y)$, then $f$ is homotopic to $g$ if and only if $f'$ is homotopic to $g'$.
Proof.
Assume first that $f$ and $g$ are homotopic, so that there exists a morphism of simplicial sets $h: \Delta ^1 \times \operatorname{Sd}(X) \rightarrow Y$ satisfying $h|_{ \{ 0\} \times \operatorname{Sd}(X) } = f$ and $h|_{ \{ 1\} \times \operatorname{Sd}(X) } = g$. The composite map
\[ \operatorname{Sd}( \Delta ^1 \times X) \rightarrow \operatorname{Sd}( \Delta ^1 ) \times \operatorname{Sd}(X) \xrightarrow { \lambda _{ \Delta ^1} \times \operatorname{id}} \Delta ^1 \times \operatorname{Sd}(X) \xrightarrow {h} Y \]
then determines a morphism of simplicial sets $h': \Delta ^1 \times X \rightarrow \operatorname{Ex}(Y)$, which is immediately seen to be a homotopy from $f'$ to $g'$.
Conversely, suppose that $f'$ and $g'$ are homotopic. Since $\operatorname{Ex}(Y)$ is a Kan complex (Corollary 3.3.5.5), we can choose a morphism of simplicial sets $h': \Delta ^1 \times X \rightarrow \operatorname{Ex}(Y)$ satisfying $h|_{ \{ 0\} \times X} = f'$ and $h'|_{ \{ 1\} \times X} = g'$, which we can identify with a map $u: \operatorname{Sd}( \Delta ^1 \times X) \rightarrow Y$. Let $v$ denote the composite map $\operatorname{Sd}( \Delta ^1 \times X) \rightarrow \operatorname{Sd}(X) \xrightarrow {f} Y$, so that $u$ and $v$ have the same restriction to $\operatorname{Sd}( \{ 0\} \times X)$. Note that the inclusion of simplicial sets $\{ 0\} \times X \hookrightarrow \Delta ^1 \times X$ is anodyne (Proposition 3.1.2.9), so the subdivision $\operatorname{Sd}( \{ 0\} \times X ) \hookrightarrow \operatorname{Sd}( \Delta ^1 \times X)$ is also anodyne (Proposition 3.3.5.3). It follows that the restriction map $\operatorname{Fun}( \operatorname{Sd}( \Delta ^1 \times X), Y) \rightarrow \operatorname{Fun}( \operatorname{Sd}( \{ 0\} \times X), Y)$ is a trivial Kan fibration, so that $u$ and $v$ belong to the same path component of $\operatorname{Fun}( \operatorname{Sd}( \Delta ^1 \times X), Y)$ and are therefore homotopic. It follows that $f = v|_{ \operatorname{Sd}( \{ 1\} \times X)}$ and $g = u|_{ \operatorname{Sd}( \{ 1\} \times X) }$ are also homotopic.
$\square$