Implication Details

Claim: If a functor is a reflector, then it is a left adjoint and is right-invertible.

Proof: If $F : \C \to \D$ is a reflector, it is left adjoint to a fully faithful functor $G : \D \to \C$. Thus, the counit $\varepsilon : F \circ G \to \id_{\D}$ is an isomorphism (Prop. 3.4 at the nLab). This shows that $G$ is a right inverse of $F$.

This implication has a dual.