Warning 10.3.3.7. The terminology of Definition 10.3.3.1 is not entirely standard. In the setting of additive categories, many authors refer to an object $Y$ as the image of a morphism $f: X \rightarrow Y$ if it is a kernel of the tautological map $Y \twoheadrightarrow \operatorname{coker}(f)$. This agrees with Definition 10.3.3.1 when $\operatorname{\mathcal{C}}$ is an abelian category (Proposition 
), but not in general.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$