= Tiny covariant functor on a Cauchy-complete category with an initial object is representable
Let $\mathcal C$ be a small <Cauchy-complete category> with <initial object> $0$. The terminal object of $[\mathcal C,\mathbf{Set}]$ is $h_0$, so $\operatorname{Nat}(P,F)\cong(F^P)(0)$. If $P$ is tiny, exponentiation by $P$ is itself a <left adjoint> and preserves all colimits. Evaluation at zero also preserves colimits. Consequently $\operatorname{Nat}(P,-)$ preserves coproducts and epimorphisms, since every epimorphism in a set-valued functor category is the coequalizer of its <kernel pair>. Thus $P$ is an <irreducible projective in a set-valued functor category>, hence <representable>.
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