Adjoint Functors Operating On Natural Transformations

Question

Suppose we have a natural transformation $\alpha$ from functor $L: \mathbb{C} \rightarrow \mathbb{D}$ to functor $L': \mathbb{C} \rightarrow \mathbb{D}$. Suppose $R: \mathbb{D} \rightarrow \mathbb{C}$ is the right adjoint of $L$. Suppose $R': \mathbb{D} \rightarrow \mathbb{C}$ is the right adjoint of $L'.$ Then is it possible to use $\alpha$ to somehow construct a natural transformation from $R'$ to $R$ ? If so how ?

Answer 1

You have, for any $x\in \mathbb D$, the map $\bar\alpha_x: R'x\to RLR'x\to RL'R'x\to Rx $ given by composing the unit of the $(L,R)$ adjunction at $R'x$ with $R(\alpha_{R'X})$ and $R$ of the counit of the $(L',R')$ adjunction at $x$. It is straightforward to check that the $\bar\alpha_x$ assemble into a natural transformation $R'\to R$, called the mate of $\alpha$ in the literature.

In fact, the mate is the key ingredient needed to construct an identity-on-objects equivalence, contravariant on 1-morphisms and 2-morphisms, between the 2-category of categories, left adjoint functors, and natural transformations between them, and the 2-category of categories, right adjoint functors, and natural transformations between them.