Constructing limits in an additive category given the existence of products and kernels

Question

The title says it all really. Given an additive category $\mathcal{A}$, is having all kernels and arbitrary products sufficient to conclude that it has all limits? Dually, is having all cokernels and arbitrary coproducts sufficient to conclude it has all colimits? I feel like this should be really quite simple, but I'm having trouble finding a reference that actually states this.

Answer 1

Yes, it is enough. Indeed, it is well-known (see for example the nLab, or Categories for the Working Mathematician, Chapter V, Section 2, Theorem 1) that a category has all limits if and only if it has arbitrary products and equalizers, so it is enough to prove that an additive category with kernels has equalizers. But given two maps $f,g:X\to Y$, for any map $h:Z\to X$ we have $fh=gh\Leftrightarrow (f-g)h=0$; thus the equalizer of $f$ and $g$ is the same thing as the kernel of $(f-g):X\to Y$.