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Adjoint functor

For adjunctions $F_i\dashv G_i$, taking mates gives a natural bijection
$$
\operatorname{Nat}(F_1,F_2)\cong\operatorname{Nat}(G_2,G_1).
$$
If $\alpha:F_1\to F_2$, its right mate is
$$
G_2\xrightarrow{\eta_1G_2}G_1F_1G_2
\xrightarrow{G_1\alpha G_2}G_1F_2G_2
\xrightarrow{G_1\varepsilon_2}G_1.
$$
Taking mates reverses vertical composition.


Mate correspondence

Monad on a left adjoint induces a comonad on its right adjoint

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 119 / 3 / Solution

Mate correspondence

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Mate correspondence

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