 Solution

Assume $\mathcal{C}$ has finite products and every idempotent morphism splits. For a representable presheaf $yA$, the exponential formula givesThus 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. Thereforepreserves 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.