Corollary 11.5.0.47. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the pullback functor
\[ U^{\ast }: (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} \rightarrow (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}} \quad \quad \operatorname{\mathcal{C}}' \mapsto \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}} \]
has a right adjoint, given on objects by the construction $\operatorname{\mathcal{E}}\mapsto \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$.