Kerodon

Remark 11.4.0.2. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets and let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of simplicial sets. Then the projection map $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ restricts to a projection map $\pi ^{\operatorname{CCart}}: \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$. Moreover, for each vertex $C \in \operatorname{\mathcal{C}}$, the isomorphism $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_ C }( \operatorname{\mathcal{D}}_ C, \operatorname{\mathcal{E}}_ C )$ of Remark 4.5.9.8 restricts to an isomorphism of full subcategories $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}) \simeq \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{D}}_ C }( \operatorname{\mathcal{D}}_ C, \operatorname{\mathcal{E}}_ C )$.