nLab adjoint triangle theorem

The adjoint triangle theorem

The adjoint triangle theorem


The adjoint triangle theorem in category theory gives conditions under which, given a pair of functors and an adjoint functor, 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) and Huq (1970).



Suppose that U:BCU \colon B\to C is a functor which has a left adjoint F:CBF \,\colon\, C\to B with the property that the diagram

FUFUϵFUFUϵFUϵ1 B F U F U \; \underoverset {\epsilon F U} {F U \epsilon} {\rightrightarrows} \; F U \xrightarrow{\epsilon} 1_B

is a pointwise coequalizer (i.e. UU is of descent type). Then for AA a category with coequalizers of reflexive pairs, a functor R:ABR \colon A\to B has a left adjoint if and only if the composite URU R does.


The direction “only if” is obvious since adjunctions compose. For “if”, let FF' be a left adjoint of URU R, and define L:BAL:B\to A to be the pointwise coequalizer of

FUFUFUϵFU F' U F U \xrightarrow{F' U \epsilon} F' U


FUFUFUθUFURFUϵFUFU F' U F U \xrightarrow{F' U \theta U} F' U R F' U \xrightarrow{\epsilon' F' U} F' U

where θ:FRF\theta:F \to R F' is the mate of the equality UR=URU R = U R under the adjunctions FUF\dashv U and FURF'\dashv U R. One then verifies that this works.


The hypotheses on UU are satisfied whenever it is monadic.


In fact, it suffices to assume that each counit ϵ:FUbb\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.


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.

It is also possible to derive the monadicity theorem from the adjoint triangle theorem Dubuc (1968).


Generalizations of the adjoint triangle theorem to 2-categories are considered in

Last revised on March 7, 2024 at 12:34:47. See the history of this page for a list of all contributions to it.