Kerodon

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

Remark 5.5.2.10. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets, let $e: X \rightarrow Y$ be an edge of $\operatorname{\mathcal{D}}$, and let $F: \operatorname{\mathcal{C}}_{X} \rightarrow \operatorname{\mathcal{C}}_{Y}$ be a functor. Then $F$ is given by covariant transport along $e$ if and only if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}_{Y}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}_{X}^{\operatorname{op}}$ is given by contravariant transport along $e$, with respect to the opposite inner fibration $q^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$.