# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 5.2.8.10 (Functoriality). Suppose we are given a commutative diagram of $\infty$-categories

5.26
$$\begin{gathered}\label{equation:enriched-homotopy-transport-functorial} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{G} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \overline{G} } & \operatorname{\mathcal{C}}', } \end{gathered}$$

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.