CatDat

Implication Details

Claim: Given a functor whose codomain is trivial, and whose domain has an initial object, then it is a coreflector.

Proof: Let $F : \C \to 1$ be the unique functor into the trivial category, and assume that $\C$ has an initial object $X$. Then the constant functor $X : 1 \to \C$ is fully faithful and left adjoint to $F$ because $\Hom(X(*),Y) \cong * \cong \Hom(*,F(Y))$.