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For a functor between Eilenberg-Moore categories lying over a right adjoint on the base categories and compatible with the monad structures, an adjoint lifting theorem constructs a left adjoint when the required reflexive coequalizers of algebra presentations exist. Applied through power-object monadicity, this converts a left adjoint of a logical functor into a right adjoint of that logical functor. The coequalizer hypotheses are part of the theorem; commutation with the forgetful functors alone is insufficient.

A logical functor between elementary toposes preserves finite limits, exponential objects and the subobject classifier, with their canonical comparison maps. It therefore preserves power objects and the double-power-object monad. If it has a left adjoint, power-object monadicity and the adjoint lifting theorem for monad algebra functors give it a right adjoint as well.