Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 5.5.2.8 (Functoriality). Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{q} \ar [r] & \operatorname{\mathcal{C}}' \ar [d]^{q'} \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{D}}', } \]

where $q'$ is a locally cartesian fibration (so that $q$ is also a locally cartesian fibration). Let $e: X \rightarrow Y$ be an edge of the simplicial set $\operatorname{\mathcal{D}}$ having image $e': X' \rightarrow Y'$ in $\operatorname{\mathcal{D}}'$. Then a functor $G: \operatorname{\mathcal{C}}_{Y} \rightarrow \operatorname{\mathcal{C}}_{X}$ is given by contravariant transport along $e$ if and only if it is given by contravariant transport along $e'$, when viewed as a functor from $\operatorname{\mathcal{C}}'_{Y'}$ to $\operatorname{\mathcal{C}}'_{X'}$.