The adjoint triangle theorem is a useful item in the categorist’s toolbox as it gives conditions under which, given a pair of functors and an adjoint, further adjoints exist.
Depending on the specific assumptions, the theorem has several variants. The following gives the most common formulation going back to Dubuc (1968).
is a coequalizer. Suppose that is a category with coequalizers of reflexive pairs; then a functor has a left adjoint if and only if the composite does.
The direction “only if” is obvious since adjunctions compose. For “if”, let be a left adjoint of , and define to be the pointwise coequalizer of
where is the mate of the equality under the adjunctions and . One then verifies that this works.
The hypotheses on are satisfied whenever it is monadic.
Similarly, the adjoint lifting theorem states conditions on a square of functors in order to ensure the existence of certain adjoints. Since a triangle can be viewed as a square with ‘two sides composed’, it is possible to deduce the adjoint lifting theorem from the adjoint triangle theorem as a corollary.
Eduardo Dubuc, Adjoint triangles, pp.69-81 in LNM 61 Springer Heidelberg 1968.
I. B. Im, G. M. Kelly, Adjoint-Triangle Theorems for Conservative Functors , Bull. Austral. Math. Soc. 36 (1987) pp.133-136.