= Solution
Assume $\mathcal C$ has finite products and every <idempotent morphism> splits. For a representable presheaf $yA$, the exponential formula gives
$$
(F^{yA})(C)
=\operatorname{Nat}(yC\times yA,F)
\cong F(C\times A).
$$
Thus exponentiation by $yA$ is precomposition with $-\times A$. Precomposition has a <Right Kan extension> as right adjoint, so every representable presheaf is a <tiny object>.
Conversely, let $P$ be tiny. Then $(-)^P$ is a left adjoint and preserves all small <colimits>. Since $\mathcal C$ has a terminal object $1$, the terminal presheaf is $y1$, and evaluation at $1$ preserves colimits. Therefore
$$
\operatorname{Nat}(P,F)
\cong\operatorname{Nat}(y1,F^P)
\cong(F^P)(1)
$$
preserves all small colimits as a functor of $F$. By the result supplied in the question, splitting idempotents implies that $P$ is representable. Hence the <representable presheaves are the tiny objects of an idempotent-complete finite-product category>, and Yoneda identifies $\mathcal C$ with the full subcategory of tiny objects of its presheaf topos.
Back to article page
