Kerodon

Notation 10.3.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$ which admits an image $Y_0 \subseteq Y$. It follows from Proposition 10.3.3.11 that the isomorphism class $[Y_0]$ is uniquely determined by $f$ (as an object of the partially ordered set $\operatorname{Sub}(Y)$; see Notation 9.3.4.27). To emphasize this, we will denote the isomorphism class $[Y_0]$ by $\operatorname{im}(f)$ and refer to it as the image of $f$. We will sometimes abuse notation by identifying $\operatorname{im}(f)$ with the object $Y_0$, viewed either as an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$ or as an object of the $\infty $-category $\operatorname{\mathcal{C}}$.