Let $H,K$ be two normal subgoups of a group $G$. We know that there exists a group isomorphism: $HK\diagup H\simeq H\diagup{H\cap K}$. I want to generalize this statement in the language of category theory.

Let $\mathscr C$ be a category with zero morphisms, kernels, cokernels, products, coproducts, equalizers and coequalizers; moreover, assume $\mathscr C$ to bo conormal. Let $h,k$ be two kernels with same codomain, say $h:H\to G$ and $k:K\to G$. Let consider the morphism $a=h\text{ coker}(k)$. We know that there exists a unique morphism $u$ such that $a=\text{ coker}(\ker (a))u\ker(\text{coker}(a))$. Moreover, by conormality, we know that $\ker(u)=0$, thus $u$ is a monomorphism.

My question: Is $u$ an isomorphism?