Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
A pair
of adjoint functors between categories $C$ and $D$, is characterized by a natural isomorphism
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 $f = g^\flat$, as in musical notation.
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.
(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
the adjunct of $g : L X \to Y$ in $C$ is the composite
Categories Work, second edition, p. 81
Category Theory in Context, p. 116, 124.
Last revised on July 21, 2017 at 14:13:57. See the history of this page for a list of all contributions to it.