Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 18 / 2 / c / Solution

A projective object in a category is an object $P$ such that, for every epimorphism $e:X\to Y$ and every $u:P\to Y$, there is $v:P\to X$ with $ev=u$.

Let $P={\coprod }_{i\in I}{P}_i$ be a coproduct in a category of projective objects in a category, with injections ${\iota }_i$. Given $e:X\to Y$ epic and $u:P\to Y$, projectivity supplies $v_i:P_i\to X$ with $e{v}_i=u{\iota }_i$. Choose these lifts for the set-indexed family. The coproduct in a category supplies a unique $v:P\to X$ satisfying $v{\iota }_i=v_i$. Since $ev{\iota }_i=u{\iota }_i$ for every $i$, its universal property gives $ev=u$.

Thus coproducts of projective objects are projective. For an empty family, $P$ is the initial object, and the lifting assertion follows directly from its unique maps. The family-of-lifts step uses the usual axiom of choice.