Kerodon

Remark 4.6.5.2. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$. For every integer $n \geq 0$, one can identify $n$-simplices of the left-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ with $(n+1)$-simplices $\sigma : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ for which $\sigma (0) = X$ and the face $d^{n+1}_0(\sigma )$ is the constant map $\Delta ^{n} \rightarrow \{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$. Similarly, one can identify $n$-simplices of the right-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ with $(n+1)$-simplices $\sigma ': \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ for which $\sigma (n+1) = Y$ and the face $d^{n+1}_{n+1}(\sigma )$ is the constant map $\Delta ^{n} \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{C}}$. In particular, we have canonical bijections