Suppose that $U:B\to C$ is a functor which has a left adjoint $F:C\to B$ with the property that the diagram
is a coequalizer. Suppose that $A$ is a category with coequalizers of reflexive pairs; then a functor $R:A\to B$ has a left adjoint if and only if the composite $U R$ does.
The direction “only if” is obvious since adjunctions compose. For “if”, let $F'$ be a left adjoint of $U R$, and define $L:B\to A$ to be the pointwise coequalizer of
and
where $\theta:F \to R F'$ is the mate of the equality $U R = U R$ under the adjunctions $F\dashv U$ and $F'\dashv U R$. One then verifies that this works.
The hypotheses on $U$ are satisfied whenever it is monadic.
In fact, it suffices to assume that each counit $\epsilon : F U b \to b$ is a regular epimorphism, rather than it is the coequalizer of a specific given pair of maps. See (Street-Verity), Lemma 2.1.
The adjoint lifting theorem is a corollary.
Eduardo Dubuc, “Adjoint triangles”, Lecture Notes in Mathematics 61
Ross Street and Dominic Verity, “The comprehensive factorization and torsors”, 2010 TAC.