# Kerodon

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Notation 5.7.8.9 (Functoriality). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Suppose we are given an arbitrary morphism of simplicial sets $f: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$, and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Precomposition with $f$ and with the projection map $\operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}$ determines a morphism of simplicial sets

$f^{\ast }: \operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}) \rightarrow \operatorname{TW}( \operatorname{\mathcal{E}}_0 /\operatorname{\mathcal{C}}_0 ),$

which we will refer to as the restriction map. Note that the construction $\operatorname{\mathcal{C}}_0 \mapsto \operatorname{TW}( \operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 )$ carries colimits in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$ to limits in the category of simplicial sets.