 A functor formula_72 from a Grothendieck categoriy formula_7 to an arbitrary category formula_74 has a left adjoint if and only if it commutes with all limits, and it has a right adjoint if and only if it commutes with all colimits.This follows from Peter J. Freyd's "special adjoint functor theorem" and its dual.The Gabriel–Popescu theorem states that any Grothendieck category formula_7 is equivalent to a full subcategory of the category formula_26 of right modules over some unital ring formula_24 (which can be taken to be the endomorphism ring of a generator of formula_7), and formula_7 can be obtained as a Gabriel quotient of formula_26 by some localizing subcategory.