Remark 4.5.9.7. Suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{U'} & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{\mathcal{B}}' \ar [r] & \operatorname{\mathcal{B}}. } \]
For every simplicial set $\operatorname{\mathcal{D}}$, we have a canonical isomorphism of simplicial sets $\operatorname{Fun}( \operatorname{\mathcal{C}}' / \operatorname{\mathcal{B}}', \operatorname{\mathcal{D}}) \simeq \operatorname{\mathcal{B}}' \times _{ \operatorname{\mathcal{B}}} \operatorname{Fun}( \operatorname{\mathcal{C}}/ \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$.