Notation 5.6.8.9 (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
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.