Definition 10.3.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $Y$ be an object of $\operatorname{\mathcal{C}}$, and let $Y_0 \subseteq Y$ be a subobject: that is, an object of $\operatorname{\mathcal{C}}$ equipped with a (specified) monomorphism $i: Y_0 \hookrightarrow Y$ (see Definition 9.3.4.26). We will say that $Y_0$ is an image of a morphism $f: X \rightarrow Y$ if the homotopy class $[f]$ factors as a composition $[i] \circ [q]$, where $q: X \twoheadrightarrow Y_0$ is a quotient morphism in $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$