Remark 8.5.1.26 (03Z5)

Remark 8.5.1.26. Let $\sigma : \Delta ^2 \twoheadrightarrow \mathcal{R}$ be the epimorphism of Notation 8.5.1.24. For every $\infty $-category $\operatorname{\mathcal{C}}$, precomposition with $\sigma $ induces a fully faithful functor $\operatorname{Fun}( \mathcal{R}, \operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$, whose essential image is the full subcategory $\operatorname{Fun}'( \Delta ^2, \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ spanned by those diagrams

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{ u} & & Y' } \]

where $u$ is an isomorphism. This follows by applying Corollary 4.5.2.29 to the pullback square

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \mathcal{R}, \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}'( \Delta ^2, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{Isom}(\operatorname{\mathcal{C}}), } \]

since the vertical maps are isofibrations (Corollary 4.4.5.3) and the lower horizontal map is an equivalence of $\infty $-categories by virtue of Corollary 4.5.3.13.