Kerodon

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Example 4.5.9.6. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, and let $U: \operatorname{\mathcal{D}}\rightarrow \Delta ^{0}$ denote the projection map. Then the direct image $\operatorname{Res}_{ \operatorname{\mathcal{D}}/ \Delta ^{0} }( \operatorname{\mathcal{E}})$ can be identified with the simplicial set

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) = \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}) } \{ \operatorname{id}_{ \operatorname{\mathcal{D}}} \} , \]

which parametrizes sections of $V$.