 Irreducible projective in a set-valued functor category

For a small category $\mathcal{C}$, an object $P$ of $[\mathcal{C},\mathbf{Set}]$ is an irreducible projective when $\mathrm{Nat}(P,-)$ preserves coproduct in a category objects and epimorphisms. Preservation of coproducts means every map from $P$ into a coproduct factors uniquely through one summand. Such objects are precisely retracts of representable functors; when the indexing category $\mathcal{C}$ is a Cauchy-complete category, they are representable. To see the retract, split the canonical epimorphism from a coproduct of representables using projectivity, and use coproduct preservation to factor the splitting through one summand.