Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker (f))\rightarrow \operatorname{Ker}(\operatorname{coker}(f))$ is both a pseudomonomorphism and a pseudoepimorphism (a pseudobimorphism). (The term derives from the fact that a preabelian category is said to be (by some---the terminology is not standard) semiabelian iff this canonical map is always a bimorphism (and of course abelian iff this canonical map is always an isomorphism), and the fact that a morphism is said to be strict iff this canonical map is an isomorphism.)
Question:
Let $\mathbf{C}$ be a finitely-complete finitely-cocomplete category with zero object and let $A_1,A_2\in \operatorname{Obj}(\mathbf{C})$. Is it necessarily the case that the projections $\pi _k\colon A_1\times A_2\rightarrow A_k$ and the inclusions $\iota _k\colon A_k\rightarrow A_1\sqcup A_2$ are semistrict?