A $1$-morphism $f\colon A\to B$ is a strict epimorphism in $\mathcal{C}$ if, for each $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the functor
\[ f^{*}\colon \mathsf{Hom}_{\mathcal{C}}(B,X)\to \mathsf{Hom}_{\mathcal{C}}(A,X) \]
given by precomposition by $f$ is injective on objects, i.e. its action on objects
\[ f_{*}\colon \operatorname {\mathrm{Obj}}(\mathsf{Hom}_{\mathcal{C}}(B,X))\to \operatorname {\mathrm{Obj}}(\mathsf{Hom}_{\mathcal{C}}(A,X)) \]
is injective.
