Remark 10.3.3.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Suppose that $\operatorname{\mathcal{C}}$ admits pullbacks, so that $f$ admits a Čechnerve $\operatorname{\check{C}}_{\bullet }( X/Y )$ (Proposition 10.2.5.6). If $f$ has an image, then $\operatorname{im}(f)$ can be identified with the geometric realization of the underlying simplicial object of $\operatorname{\check{C}}_{\bullet }(X/Y)$. To see this, choose a $2$-simplex
which exhibits $Y_0$ as an image of $f$. Since $i$ is a monomorphism, Remark 10.3.1.17 supplies an isomorphism between the underlying simplicial objects of $\operatorname{\check{C}}_{\bullet }(X/Y)$ and $\operatorname{\check{C}}_{\bullet }(X/Y_0)$. It will therefore suffice to show that $\operatorname{\check{C}}_{\bullet }(X/Y_0)$ is a colimit diagram in $\operatorname{\mathcal{C}}$, which is a reformulation of our assumption that $q$ is a quotient morphism (Proposition 10.3.2.4).