CatDat

Implication Details

Claim: Given a functor whose domain has coproducts and is locally essentially small, if it is representable, then it is a right adjoint.

Proof: If $\C$ is a locally small category with coproducts and $X \in \C$ is any object, then the copower functor $\Set \to \C$, $T \mapsto T \otimes X$ is left adjoint to $\Hom(X,-) : \C \to \Set$.