Let $\mathcal{C}$ be an abelian category. Consider the following commutative diagram:
I am trying to prove the following version of the four lemma:
if $\alpha$ is an epimorphism and $\beta$ and $\gamma$ are monomorphisms, then $\mu$ is a monomorphism.
My solution thus far is as follows. Consider the following augmented diagram:
The horizontal arrows in the top row are the unique zero morphisms. Note that: $$\gamma \circ h \circ \ker \mu = h' \circ \mu \circ \ker \mu = h' \circ 0 = 0$$ Also, $$ h \circ \ker \mu = 0 $$ Since $g$ is the kernel of $h$, there exists a unique morphism $\varphi : \ker \mu \to B$. Note that $$ g' \circ ( \beta \circ \varphi ) = \mu \circ g \circ \varphi = 0 $$ Since $f'$ is the kernel of $g'$, there exists a unique morphism $\psi_1 : \ker \mu \to A'$.
I am unable to proceed beyong this point. The only remaining step is to essentially list the morphism $\ker \mu \to A'$ to a morphism $\ker \mu \to A$, but I cannot complete this argument using only universal property of the cokernel.
Can anyone help?