From Algebra: Chapter $0$ by Aluffi:
He proves this by showing $N \cong M \oplus \ker \psi$:
However, as I read Hatcher's Algebraic Topology this past semester, he has us show (in this case) $N \cong M \oplus \mathrm{coker} \varphi$:
I managed to put together the maps used by Aluffi and by Hatcher to come up with the isomorphism $M \oplus \ker \psi \cong M \oplus \mathrm{coker} \varphi$.
However, as Hatcher's proof seemed more intuitive, I was wondering why Aluffi chose to show $N \cong M \oplus \ker \psi$? It also doesn't seem like he is using $\mathrm{coker}\varphi$ in the proof, so I was extra puzzled.