Warning 10.3.3.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $C_{\bullet }$ denote the underlying simplicial object of the Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$. Remark 10.3.3.15 asserts that if $f$ has an image, then that image can be identified with a geometric realization of $C_{\bullet }$. Beware that the converse is false in general. Suppose that $C_{\bullet }$ admits a geometric realization $| C_{\bullet } |$, given by the image on an initial object $E$ of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{ C_{\bullet } / }$. The augmented simplicial object $\operatorname{\check{C}}_{\bullet }(X/Y)$ determines another object $\widetilde{Y} \in \operatorname{\mathcal{C}}_{ C_{\bullet } / }$, so there is an (essentially unique) morphism from $E$ to $\widetilde{Y}$. The forgetful functor $\operatorname{\mathcal{C}}_{C_{\bullet } / } \rightarrow \operatorname{\mathcal{C}}_{X/}$ carries this morphism to a $2$-simplex
in the $\infty $-category $\operatorname{\mathcal{C}}$. In this situation, the following conditions are equivalent:
The morphism $i$ is a monomorphism.
The diagram (10.25) exhibits $| C_{\bullet } |$ as an image of $f$.
The morphism $f$ has an image in $\operatorname{\mathcal{C}}$.