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