nLab
opposite adjunction
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Category theory
category theory

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Idea
Given a pair of adjoint functors

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

there is induced an 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:

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

Last revised on July 20, 2021 at 09:55:41.
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