Definition 2.10 (Projective). We say an object $P$ in a locally small category $C$is projective if $C(P,\bullet )$ preserves epimorphisms, i.e. if given

there exists $h:P\to Q$ with $gh=f$. Dually, $P$ is injective if it’s projective in $C^{op}$.

If $P$ satisfies this condition for all $g$ in some class $E$ of epimorphisms, we call it $E$-projective.