= Solution
{c}
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> give
$$
\operatorname{Nat}(P,F)\cong\operatorname{Nat}(h_0,F^P)\cong(F^P)(0).
$$
Since $P$ 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 $\operatorname{Nat}(P,-)$ 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:
$$
\boxed{P\text{ tiny}\ \Longrightarrow\ P\cong\mathcal C(A,-)\text{ for some }A.}
$$
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