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

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, transpose, and mate.

Properties

Proposition

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

Let $i_X : X \to R L X$ be the unit of the adjunction and $\eta_X : L R X \to X$ the counit.

Then

the adjunct of $f : X \to R Y$ in $D$ is the composite

$\tilde f : L X \stackrel{L f}{\to} L R Y \stackrel{\eta_Y}{\to} Y$

the adjunct of $g : L X \to Y$ in $C$ is the composite

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

Examples

The process of currying is an instance of passage to adjuncts, specialized to the tensor-hom adjunction of a closed monoidal category.