Kerodon

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

Proposition 4.6.5.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ are Kan complexes.

Proof. By virtue of Proposition 4.3.6.1, the projection map $\operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration. Applying Corollary 4.4.2.3, we deduce that the fiber $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) = \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Y\} $ is a Kan complex. A similar argument shows that $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ is a Kan complex. $\square$