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Notation 5.3.7.8 (Cocartesian Direct Images). Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be morphisms of simplicial sets, and let $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})$ be the simplicial set of Construction 4.5.9.1. Using Remark 4.5.9.8, we can identify vertices of the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})$ with pairs $(C, F)$, where $C$ is a vertex of $\operatorname{\mathcal{C}}$ and

$F: \operatorname{\mathcal{D}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}= \operatorname{\mathcal{E}}_{C}$

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 $U$.