Remark 4.5.9.8. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets, let $\operatorname{\mathcal{D}}$ be a simplicial set, and let $\pi : \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ be the projection map of Construction 4.5.9.1. For every vertex $B \in \operatorname{\mathcal{B}}$, Remark 4.5.9.7 and Example 4.5.9.2 supply an isomorphism of simplicial sets
\[ \pi ^{-1} \{ B\} = \{ B \} \times _{\operatorname{\mathcal{B}}} \operatorname{Fun}( \operatorname{\mathcal{C}}/ \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}( \{ B\} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}). \]