Proposition 4.5.9.5 (02TP)

Proposition 4.5.9.5. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets and let $\operatorname{\mathcal{D}}$ be a simplicial set. For every morphism of simplicial sets $\operatorname{\mathcal{B}}' \rightarrow \operatorname{\mathcal{B}}$, postcomposition with the evaluation map $\operatorname{ev}: \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{B}}} \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{D}}$ induces a bijection

\[ \operatorname{Hom}_{ (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{B}}} }( \operatorname{\mathcal{B}}', \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{B}}', \operatorname{\mathcal{D}}). \]

Proof. Writing $\operatorname{\mathcal{B}}'$ as a colimit of simplices, we may reduce to the case where $\operatorname{\mathcal{B}}' = \Delta ^{n}$, so that $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{B}}$. In this case, the desired result follows immediately from the definition of the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$. $\square$