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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.

Let $P$ be a tiny object of $[\mathcal{C},\mathbf{Set}]$, and let $0$ be the initial object of $\mathcal{C}$. The representable functor $h_0=\mathcal{C}(0,-)$ is the terminal functor, since each of its values is a singleton. The exponential adjunction and the Yoneda lemma giveSince $P$ is tiny, $E_P:F\mapsto F^P$ is itself a left adjoint, so preserves all colimits. Evaluation at $0$ also preserves colimits, because they are pointwise in a set-valued functor category. Therefore $\mathrm{Nat}(P,-)$ preserves the coproduct construction.

It also preserves epimorphisms. Indeed, every such epimorphism is a pointwise epimorphism in a functor category, hence the coequalizer of its kernel pair, as can be checked pointwise in sets. A left adjoint preserves that coequalizer and therefore sends it to a regular epimorphism, which is an epimorphism; evaluation at zero preserves surjectivity. Thus $P$ is an irreducible projective in a set-valued functor category.

A Cauchy-complete category is one in which every idempotent morphism splits. By the characterization allowed in the question, irreducible projectives are representable functors for such an indexing category. Applying it to $P$ proves the tiny covariant functor on a Cauchy-complete category with an initial object is representable conclusion:

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.