= Projective cover of a set-valued functor by representables
For a <small category> $\mathcal C$ and $F:\mathcal C\to\mathbf{Set}$, the <Yoneda lemma> gives a canonical pointwise-surjective natural transformation
$$
\coprod_{(A,x),\ x\in F(A)}\mathcal C(A,-)\longrightarrow F.
$$
Each representable is projective because evaluation preserves pointwise epimorphisms, and their coproduct is projective. Thus every set-valued functor on a small category is an epimorphic image of a projective object.
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