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Let $T$ and $S$ be monads on categories $\mathcal{C}$ and $\mathcal{D}$, with forgetful functors $U_T,U_S$. Suppose $K:{\mathcal{D}}^S\to {\mathcal{C}}^T$ and $U:\mathcal{D}\to \mathcal{C}$ satisfy $U_{T}K=U{U}_S$. If $F⊣U$ and ${\mathcal{D}}^S$ has coequalizers of reflexive pairs, then $K$ has a left adjoint $L$. Thus the square must specify an actual lifting to Eilenberg-Moore categories; the required coequalizers are taken in the category of algebras.^[ref]

Writing $F_T,F_S$ for the free algebra functors, the construction starts withbecause the adjunctions give natural bijectionsAn algebra for a monad $(A,a)$ has the reflexive free-algebra presentation of a monad algebra $F_T(TA)\rightrightarrows F_{T}A\to (A,a)$. Transporting its two arrows through these bijections gives a reflexive pair between $F_{S}F(TA)$ and $F_{S}FA$; its coequalizer defines $L(A,a)$. The universal property supplies $L⊣K$.

This explains the theorem’s use in constructing adjoints of logical functors through power-object monadicity: the algebra presentation makes the missing adjoint explicit.

Jaap van Oosten works on category theory, the theory of the topos and realizability, connecting mathematical structures with models of computation. Utrecht's publication record verifies this research program.^[ref]

His Topos Theory by Jaap van Oosten (2018) explains how categorical structure can transport logical constructions. Its adjoint lifting theorem for monad algebra functors builds an adjoint at an algebraic level from an adjunction and suitable presentations underneath. The notes apply this to logical functors using power-object structure, making precise when a familiar construction survives passage to a richer category.^[ref]