 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 $\mathrm{Nat}(P,F)\cong (F^P)(0)$. If $P$ is tiny, exponentiation by $P$ is itself a left adjoint and preserves all colimits. Evaluation at zero also preserves colimits. Consequently $\mathrm{Nat}(P,-)$ 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.