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