Representable presheaves are the tiny objects of an idempotent-complete finite-product category

If a small finite-product category $\mathcal{C}$ splits idempotents, the tiny objects of its presheaf topos are exactly the representable presheaves. Exponentiation by $yA$ is precomposition with $-\times A$ and therefore has a right adjoint given by Right Kan extension. Conversely, if $P$ is tiny, then $\mathrm{Nat}(P,-)$ preserves all colimits; idempotent completeness makes every such presheaf representable.