= 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 $\operatorname{Nat}(P,-)$ 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.
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