Remark 10.3.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a $2$-simplex
where $g$ is a monomorphism. Suppose that $\operatorname{\mathcal{C}}$ admits pullbacks, so that the morphisms $f$ and $h$ admit Čechnerves $\operatorname{\check{C}}_{\bullet }(X/Y)$ and $\operatorname{\check{C}}_{\bullet }(X/Z)$ (Proposition 10.2.5.6). Then the underlying simplicial objects of $\operatorname{\check{C}}_{\bullet }(X/Y)$ and $\operatorname{\check{C}}_{\bullet }(X/Z)$ are canonically isomorphic. To see this, let us regard (10.19) as morphism $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ in the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Z}$. Since the forgetful functor $\operatorname{\mathcal{C}}_{/Z} \rightarrow \operatorname{\mathcal{C}}$ preserves pullbacks (Corollary 7.1.6.20), we can identify $\operatorname{\check{C}}_{\bullet }(X/Y)$ with the image of $\operatorname{\check{C}}_{\bullet }( \widetilde{X} / \widetilde{Y} )$. The desired result now follows by applying Example 10.3.1.16 to the object $\widetilde{Y} \in \operatorname{\mathcal{C}}_{/Z}$ (which is subterminal by virtue of Remark 9.3.4.16).