Notation 5.3.7.8 (Cocartesian Direct Images). Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets $\operatorname{id}_{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ determines a section of the projection map $\pi : \operatorname{Fun}( \operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$. For every morphism of simplicial sets $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$, we let $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ denote the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) } \operatorname{Fun}(\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. Unwinding the definitions, we see that vertices of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ can be identified with pairs $(C,F)$, where $C$ is a vertex of $\operatorname{\mathcal{C}}$ and
is a section of the map $V|_{\operatorname{\mathcal{E}}_ C}: \operatorname{\mathcal{E}}_ C \rightarrow \operatorname{\mathcal{D}}_{C}$. If $V$ is a cocartesian fibration, we let $\operatorname{Res}^{\operatorname{CCart}}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})$ denote the full simplicial subset of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ spanned by those vertices $(C,F)$ where $F$ carries each edge of $\operatorname{\mathcal{D}}_{C}$ to $V_{C}$-cocartesian edge of $\operatorname{\mathcal{E}}_{C}$. We will refer to $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}})$ as the cocartesian direct image of $\operatorname{\mathcal{E}}$ along $U$.