Kerodon

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

Proposition 4.6.1.19. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $e: q(X) \rightarrow q(Y) )$ be a morphism in $\operatorname{\mathcal{D}}$. Then the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( q(X), q(Y) )$ is a Kan fibration of simplicial sets.

Proof of Proposition 4.6.1.19. Apply Proposition 4.6.1.22 in the special case $B = \Delta ^1$ and $A = \operatorname{\partial \Delta }^1$. $\square$