I want to show that the functor category $A^J$ is an abelian category if $A$ is an abelian category. I know it's easy to define a null object, binary biproducts, kernels, and cokernels.
But I got stuck on the last condition that every monomorphism is a kernel, and every epimorphism is a cokernel.
Suppose that $\alpha \colon S \to T$ is a natural transformation, and it's a monomorphism in $A^J$, but I don't think we can conclude that $\alpha_j \colon S_j \to T_j$ is a monomorphism in $A$. Then it's difficult to verify the last condition by universal properties.
My solution I think we can conclude $\ker \alpha=0$ if $\alpha$ is monic. Since if $\alpha \circ \beta =0$, then $\beta = 0$ ($\alpha$ is monic) factors through $\ker \alpha =0$. So it's obvious that the last condition holds for $A^J$.
Could you tell me whether my solution is right? Thank you.