Remark 5.2.8.10 (Functoriality). Suppose we are given a commutative diagram of $\infty $-categories
where $U$ and $U'$ are cocartesian fibrations and the functor $G$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ to $U'$-cocartesian morphisms of $\operatorname{\mathcal{E}}'$. For each object $C \in \operatorname{\mathcal{C}}$ having image $C' = \overline{G}(C)$, $G$ restricts to a functor $G_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C'}$. It follows from Remark 5.2.8.5 that the construction $C \mapsto G_{C}$ determines a natural transformation of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors $\alpha : \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \rightarrow \operatorname{hTr}_{\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}'} \circ \mathrm{h} \mathit{\overline{G}}$ from $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to $\mathrm{h} \mathit{\operatorname{QCat}}$. Moreover, if (5.26) is a pullback square, then $\alpha $ is an isomorphism of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors.