# Kerodon

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Remark 5.3.1.17 (Functoriality). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. Suppose that we are given a morphism of simplicial sets $F: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$, and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}'_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$. Then pullback along $F$ determines a morphism of simplicial sets

$F^{\ast }: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }^{\operatorname{CCart}}( \operatorname{\mathcal{E}}'_0, \operatorname{\mathcal{E}}_0 ),$

which we will refer to as the restriction map.