= 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 $\operatorname{Nat}(P,-)$ preserves all colimits; idempotent completeness makes every such presheaf representable.
Back to article page
