= 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 $ev_i=u\iota_i$. 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 \b[<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>.
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