Construction 1.5.3.1. Let $S$ and $T$ be simplicial sets. Then the construction
\[ ([n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n \times S, T ) \]
determines a functor from the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the category of sets. We regard this functor as a simplicial set which we will denote by $\operatorname{Fun}( S, T)$.
Note that, given an $n$-simplex $f$ of $\operatorname{Fun}( S, T )$ and an $n$-simplex $\sigma $ of $S$, we can construct an $n$-simplex $\operatorname{ev}( f, \sigma )$ of $T$, given by the composition
\[ \Delta ^ n \xrightarrow {\delta } \Delta ^ n \times \Delta ^ n \xrightarrow { \operatorname{id}\times \sigma } \Delta ^ n \times S \xrightarrow { f } T. \]
This construction determines a map of simplicial sets $\operatorname{ev}: \operatorname{Fun}( S, T) \times { S} \rightarrow T$, which we will refer to as the evaluation map.