The Yoneda Ext and arbitrary coproducts in abelian categories

Abstract: There are well known identities that involve the Ext bifunctor, coproducts, and products in Ab4 and Ab4* abelian categories with enough projectives and enough injectives. Namely, for every such category $\mathcal{A}$, the isomorphisms $$\operatorname{Ext}ⁿ (\bigoplus_{i\in I}A_{i},X) \cong \prod_{i\in I} \operatorname{Ext}<sup>n(A_{i},X)$$ and $$\operatorname{Ext}ⁿ (X,\prod_{i\in I}A_{i}) \cong \prod_{i\in I}\operatorname{Ext}ⁿ (X,A_{i})$$ always exist. The goal of this paper is to show similar isomorphisms for the Yoneda Ext in Ab4 and Ab4* abelian categories with not necessarily enough projectives nor injectives. The desired isomorphisms are constructed explicitely by using limits and colimits.