On page 55 of Weibel's Introduction to homological algebra the following passage appears:

Here are two consequences that use the fact that homology commutes with arbitrary direct sums of chain complexes

I understand why homology commutes with arbitrary direct sums when the direct sum of a collection of monics is a monic (i.e the direct sum functor is exact) but I was under the impression that there were abelian categories where the direct sum functor is not exact. After a bit of thought, I realised that I don't know an example of an abelian category in which the coproduct functor is not exact.

Sheaves of abelian groups on a fixed topological space give an example of an abelian category in which the product functor is not exact.

Question 1: Is the passage from Weibel's book correct? If so, then why?

Question 2: Is there an example of an abelian category where the direct sum functor is not exact?