 Coproducts of projective objects are projective

A set-indexed coproduct in a category of projective objects in a category is projective. Restrict a map out of the coproduct to each summand, lift each restriction through the given epimorphism, and assemble the lifts by the coproduct universal property. Choosing an arbitrary family of lifts uses the usual axiom of choice. The empty coproduct, an initial object, is projective as well.