nLab
adjunct

Definition

A pair

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

of adjoint functors between categories $C$ and $D$, is characterized by a natural isomorphism

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

of hom-sets for objects $X\in D$ and $Y\in C$. Two morphisms $f:LX\to Y$ and $\tilde{f}:X\to RY$ which correspond under this bijection are said to be adjuncts of each other. That is, $\tilde{f}$ is the adjunct of $f$ and $f$ is the adjunct of $\tilde{f}$.

Sometimes people call $\tilde{f}$ the “adjoint” of $f$, and vice versa, but this is potentially confusing because it is the functors $F$ and $G$ which are adjoint. Other possible terms are conjugate and mate.

Properties

Let $i_X:X\to RLX$ be the unit of the adjunction and ${\eta }_X:LRX\to X$ the counit.

Then

• the adjunct of $f:X\to RY$ in $D$ is the composite

  $$\tilde{f}:LX\stackrel{Lf}{\to }LRY\stackrel{{\eta }_Y}{\to }Y$$\tilde f : L X \stackrel{L f}{\to} L R Y \stackrel{\eta_Y}{\to} Y

• the adjunct of $g:LX\to Y$ in $C$ is the composite

  $$\tilde{g}:X\stackrel{i_X}{\to }RLX\stackrel{Rg}{\to }RY$$\tilde g : X \stackrel{i_X}{\to} R L X \stackrel{R g}{\to} R Y

Related concepts

• adjunction

• zig-zag law/triangle identity

• unit of an adjunction

• adjunct