$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 1.5.3.2. Let $S$, $T$, and $U$ be simplicial sets. Then the composite map
\begin{eqnarray*} \theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( U, \operatorname{Fun}( S, T ) ) & \rightarrow & \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( U \times S, \operatorname{Fun}( S, T ) \times S )\\ & \xrightarrow { \operatorname{ev}\circ } & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( U \times S, T ) \end{eqnarray*}
is bijective.
Proof.
Let $f: U \times S \rightarrow T$ be a map of simplicial sets. For each $n$-simplex $\sigma $ of $U$, the composite map
\[ \Delta ^ n \times S \xrightarrow { \sigma \times \operatorname{id}} U \times S \xrightarrow {f} T \]
can be regarded as an $n$-simplex of $\operatorname{Fun}( S, T)$, which we will denote by $g(\sigma )$. The construction $\sigma \mapsto g(\sigma )$ determines a map of simplicial sets $g: U \rightarrow \operatorname{Fun}( S, T )$. We leave as an exercise for the reader to verify that $g$ is the unique map satisfying $\theta (g) = f$.
$\square$