nLab opposite adjunction

Idea

Given a pair of adjoint functors

$$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} \mathcal{C}$$

there is an induced adjunction of opposite functors between their opposite categories of the form

$$\mathcal{C}^{op} \underoverset {\underset{L^{op}}{\longrightarrow}} {\overset{R^{op}}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} \mathcal{D}^{op} \,.$$

Hence where $L$ was the left adjoint, its opposite becomes the right adjoint, and dually for $R$.

This is immediate from the definition of opposite categories and the characterization of adjoint functors via the corresponding hom-isomorphism.

The adjunction unit of the opposite adjunction has as components the components of the original adjunction counit, regarded in the opposite category, and dually:

$$\epsilon^{R^{op} L^{op}}_{d} \;\colon\; R^{op}\circ L^{op}(d) \xrightarrow{\;\; \big( \eta^{R L}_d \big)^{op} \;\;} d \,, {\phantom{AAAAAA}} \eta^{L^{op} R^{op}}_{c} \;\colon\; c \xrightarrow{\;\; \big( \epsilon^{L R}_c \big)^{op} \;\;} L^{op} \circ R^{op}(c) \,.$$

Related concepts

• opposite Quillen adjunction

• opposite category, opposite functor

• adjunction

• opposite 2-category