Implication Details
Claim: Given a functor whose codomain is locally essentially small, and whose domain has a cogenerating set and is complete and is locally essentially small and is well-powered, if it is continuous, then it is a right adjoint.
Proof: This is the Special Adjoint Functor Theorem. The proof can be found, for example, at the nLab, or in Mac Lane, Ch. V, Theorem 8.2.