Kerodon

Proof. Let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$, which we view as a morphism of simplicial sets $\tau : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ whose restriction to the face $\Delta ^{n} \subseteq \Delta ^{n+1}$ equal to the constant map $\Delta ^{n} \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{C}}$. Then $\sigma $ belongs to the simplicial subset $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y) \subseteq \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$ if and only if, for every $2$-simplex $\rho : \Delta ^2 \rightarrow \Delta ^{n+1}$, the composition $\tau \circ \rho $ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$. Note that this condition is automatically satisfied if $\rho $ is degenerate, or takes values in the subset $\Delta ^{n} \subseteq \Delta ^{n+1}$ (since every degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ is thin). Consequently, it suffices to verify this condition in the case where $\rho $ is the right cone of a map $\rho _0: \Delta ^1 \rightarrow \Delta ^{n}$. In this case, $\tau \circ \rho $ is thin if and only if the edge $\Delta ^{1} \xrightarrow {\rho _0} \Delta ^{n} \xrightarrow {\sigma } \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$ is an isomorphism in the $\infty $-category $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$ (Theorem 5.4.4.1). Allowing $\tau _0$ to vary, we obtain the identification $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y) \simeq \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)^{\simeq }$; the proof of the analogous statement for left-pinched morphism spaces is similar. $\square$