Kerodon

Remark 5.4.3.7. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category containing $X$ and $Y$. We will see later that the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}( X, Y)$ is naturally equivalent to the opposite of the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( X, Y)$ (Proposition ). When $\operatorname{\mathcal{C}}$ is the Duskin nerve of a $2$-category, we can do better: the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}( X, Y)$ is isomorphic to the opposite of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$; see Example 4.6.5.13.