Additive functors from a small preadditive category to an abelian category form an abelian category

By [L1], the additive functors already form a preadditive category. The zero functor is additive, and binary biproducts are computed pointwise because [L4] gives biproducts in $\mathcal{A}$ and [L3] computes them pointwise. So the additive functor category is additive.

Let $\eta :F\Rightarrow G$ be a natural transformation. By [L3], its kernel and cokernel in the ambient functor category are computed pointwise, and the pointwise constructions lie in $\mathcal{A}$. Because $\mathcal{C}$ is preadditive and $F,G$ are additive, the induced structure maps on those pointwise kernels and cokernels are again additive by uniqueness in the kernel and cokernel universal properties. The canonical coimage-to-image map is likewise computed pointwise and is an isomorphism at each object by [L4]. Therefore the additive functor category satisfies the axioms of an abelian category.