If $A$ abelian category and $X$ a small category (a set) then $Fun(X,A)$ is abelian category.

Question

I managed to show that $\operatorname{Fun}(X,A)=\{\text{functors}:X\to A\}$ is a category, and I am trying to show that it is Abelian. It is easy to show that is pre-additive, but I can't define the zero object of the category, to prove that every homomorphism has a kernel and cokernel and finally that the induced homomorphism $f^{-}\colon\text{CoIm}f\to\text{Im}f$ is isomorphism for every $f$ homomorphism. I actually have a problem on how to use the zero object of $A$ to define the zero object of $\operatorname{Fun}(X,A)$. I would appreciate any help.