Corollary 8.5.1.30. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then composition with the retraction diagram of Notation 8.5.1.24 induces a fully faithful functor $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$, whose essential image is 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.