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 $\eta_X \colon X \to R L X$ be the unit of the adjunction and $\epsilon_X \colon L R X \to X$ the counit.
Then
the adjunct of $f \colon X \to R Y$ in $D$ is the composite
the adjunct of $g \colon L X \to Y$ in $C$ is the composite
For proof see this Prop. at adjoint functor.
Categories Work, second edition, p. 81
Category Theory in Context, p. 116, 124.
Last revised on May 31, 2023 at 07:23:38. See the history of this page for a list of all contributions to it.