nLab adjunct




A pair

(LR):CRLD (L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D

of adjoint functors between categories CC and DD, is characterized by a natural isomorphism

C(LX,Y)D(X,RY) C(L X,Y) \cong D(X,R Y)

of hom-sets for objects XDX\in D and YCY\in C. Two morphisms f:LXYf:L X \to Y and g:XRYg : X \to R Y which correspond under this bijection are said to be adjuncts of each other. That is, gg is the (right-)adjunct of ff, and ff is the (left-)adjunct of gg. Sometimes one writes g=f g = f^\sharp and f=g f = g^\flat, as in musical notation.

Sometimes people call f˜\tilde f the “adjoint” of ff, and vice versa, but this is potentially confusing because it is the functors FF and GG which are adjoint. Other possible terms are conjugate, transpose, and mate.



(adjuncts in terms of adjunction (co-)unit)

Let η X:XRLX\eta_X \colon X \to R L X be the unit of the adjunction and ϵ X:LRXX\epsilon_X \colon L R X \to X the counit.


  • the adjunct of f:XRYf \colon X \to R Y in DD is the composite

    f˜:LXLfLRYϵ YY \tilde f \colon L X \overset{L f}{\longrightarrow} L R Y \overset{\epsilon_Y}{\longrightarrow} Y
  • the adjunct of g:LXYg \colon L X \to Y in CC is the composite

    g˜:Xη XRLXRgRY. \tilde g \colon X \stackrel{\eta_X}{\longrightarrow} R L X \overset{R g}{\longrightarrow} R Y \,.

For proof see this Prop. at adjoint functor.



Last revised on May 31, 2023 at 07:23:38. See the history of this page for a list of all contributions to it.