The category of countable abelian groups is an essentially small abelian category, and has enough projectives and injectives (the countable free abelian groups and the countable divisible groups respectively). However, there is an injective with endomorphism ring $\mathbb{Q}$, but no such projective, so the categories of projectives and injectives can't be dual.

The category of countable abelian groups is an essentially small abelian category, and has enough projectives and injectives (the countable free abelian groups and the countable divisible groups respectively). However, there is an injective with endomorphism ring $\mathbb{Q}$, but no such projective, so the categories of projectives and injectives can't be equivalent or dual.