Proposition 10.3.3.6 (Images of Quotient Morphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is a quotient morphism if and only if $Y$ is an image of $f$ (when regarded as a subobject of itself).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Assume first that $f$ is a quotient morphism. Since the identity map $\operatorname{id}_{Y}$ is a monomorphism (Example 9.3.4.9), the right-degenerate $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{\operatorname{id}_ Y} & \\ X \ar [ur]^{f} \ar [rr]^{f} & & Y } \]
exhibits $Y$ as an image of $f$. Conversely, if $Y$ is an image of $f$, then $f$ factors as the composition of a quotient morphism and an isomorphism, and is therefore a quotient morphism by virtue of Corollary 10.3.2.12. $\square$