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 be 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?

In order to prove it, it's enoght to show that the image of $a$ is normal, that's a kernel.
In particular, consider the following diagram:
\begin{matrix}
           & \rightarrow & K          &             &            \cr
\downarrow &             & \downarrow &             &            \cr
H          & \rightarrow & G          & \rightarrow & G\diagup H \cr
           &            & \downarrow &             & \downarrow \cr
           &             & G\diagup K & \rightarrow_q &            
\end{matrix}
in which the upper square is a pullback while the other below is a pushfoward.
Then it's enoght to show that the image of $a$ is the kernel of $q$.