Let $\mathsf{C}$ be a category. We call $\mathsf{C}$ binormal if it has a null object, has all equilizers and coequilizers, all monomorphisms are kernels and all epimorphisms are cokernels (whereby a kernel means an equalizer of a morphism and a zero morphism, and cokernel dually). Then my question is:

Can we make any binormal category a preadditive category?

Here I'm using wikipedia's definition of preadditive category.

I know that a binormal category which has all binary products must be abelian. I'm just wondering if we can drop out the requirement of the existence of products.