# Kerodon

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

Corollary 5.4.3.5. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. For every pair of objects $X$ and $Y$, the pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ of Construction 4.6.5.1 are $\infty$-categories.

Proof. By definition, the left-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ is the fiber over $Y$ of the projection map $\pi : \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$. Since $\pi$ is an interior fibration (Proposition 5.4.3.1), each of its fibers is an $\infty$-category (Remark 5.4.2.5). A similar argument shows that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ is an $\infty$-category. $\square$