Let $C, D$ be abelian categories. Let $F : C → D$ and $G : D → C$ be additive covariant functors. Suppose that $G$ is left adjoint to $F$. Then $G$ is right exact.
In the proof, let $U → V → W → 0$ be an exact sequence in $D$, thus we have an exact sequence $0 \to Hom_C(G(W), X ) \to Hom_C(G(V), X ) \to Hom_C(G(U), X)$. Out aim is to show the exactness of $G(U) \to G(V) \to G(W) \to 0$. The author states that one shows that a cokernel $G(V) → Y$ of the morphism $G(U) → G(V)$ factors through $G(W)$, from which one deduces the exactness at $G(V)$.
My question is that is factoring through $G(W)$ sufficient, I tried this in Mod, the conclusion didn't follow. Any help would be appreciated
Exactness of $G(U)\to G(V)\to G(W)\to0$ means that $G(V)\to G(W)$ is the cokernel of $G(U)\to G(V)$, and $G(V)\to G(W)$ is an epimorphism. Since $G$ is a left adjoint, these both follow from the fact that left adjoints preserve colimits.
To be more specific about your question, recall that the cokernel of $G(U)\to G(V)$ is its coequaliser with the zero morphism, meaning that it is a morphism $G(V)\to K$ such that $G(U)\to G(V)\to K$ is zero and whenever $G(U)\to G(V)\to Y$ is zero, $G(V)\to Y$ must factor uniquely through $G(V)\to K$. Note that since $U\to V\to W$ is zero, the same holds for $G(U)\to G(V)\to G(W)$ (either using that $G$ is additive, or just the fact that $G$ preserves colimits will do). If any cokernel factors uniquely* through $G(W)$, this means that $G(W)$ has the universal property of the cokernel and thus must be the cokernel. I suppose this is "all you have to do" because the argument is analogous for showing that $G(V)\to G(W)$ is an epimorphism, since this amounts to showing that $G(W)\to 0=G(0)$ is its cokernel.
*uniqueness is important; without it, $G(W)$ will not necessarily be the cokernel.
To see this is indeed true, note that morphisms $G(V)\to K$ correspond naturally with morphisms $V\to F(K)$ by the adjointness, and zero morphisms are preserved under this correspondence (because $G(V)\to0$ corresponds to $V\to F(0)=0$ and $0=G(0)\to Y$ corresponds to $0\to F(Y)$). In particular, $G(U)\to G(V)\to K$ being zero means the same for $U\to V\to F(K)$. As $W$ is the cokernel of $U\to V$, there is a unique factoring morphism $W\to F(K)$ and by the adjointness this corresponds to a unique factoring morphism $G(W)\to K$.