What is the meaning of $F\eta$ in the definition of adjunction

Question

I'm rather new at this so please try to make it simple.

Reading the wikipedia's adjunction Definition via counit–unit in the following link: https://en.wikipedia.org/wiki/Adjoint_functors#Definition_via_counit%E2%80%93unit_adjunction, I came across the symbol $F\eta$, where $\eta:1_{\cal D}\rightarrow GF$ and $F:\cal D\rightarrow\cal C$ and $G:\cal C\rightarrow \cal D$; $\eta$ is a natural transformation, $F,G$ are functors and $\cal C, \cal D$ are categories.

This is probably a silly question but what does $F\eta$ represents? is it $F\circ\eta$?
In that case, how does it work? in the diagram it says: $F\eta(F)\rightarrow FGF$, but $\eta$'s domain does not contain $F$..
Clarification would be much appreciated.

Answer 1

First, notice that Wikipedia has $F : \mathcal D \to \mathcal C$ and $G : \mathcal C \to \mathcal D$, rather than the other way around.

$F \eta : F \Rightarrow F G F$ is the whiskering of $\eta$ with $F$. This is described in the last two paragraphs here. Whiskering can be seen as a special case of composition of natural transformations, if we identify the functor $F$ with the identity natural transformation on $F$.

Recall that $\eta$ is a family of morphisms $\eta_D : D \to GF(D)$. $F \eta$ is then a natural transformation constructed from $F$ and $\eta$, defined by $(F\eta)_D = F(\eta_D) : F(D) \to FGF(D)$.

There is an analogous whiskering $\epsilon F : F G F \Rightarrow F$, of $F$ with $\epsilon$, given by $(\epsilon F)_D = \epsilon_{F(D)} : FGF(D) \to F(D)$. Similarly, we can whisker $G$ with both natural transformations as well.

Answer 2

The general construction is that of whiskering or horizontal composition of a 2-morphism (natural transformation) with a 1-morphism (functor). In general:

Let $F,G: \mathcal{C} \rightarrow \mathcal{D}$ be two functors, $H: \mathcal{D} \rightarrow \mathcal{E}$, and $\eta: F \Rightarrow G$, then one may define the following natural transformation: $H \eta: HF \Rightarrow HG$ via $(H\eta)_x: HF(x) \rightarrow HG(x)$ defined by applying $H$ to $\eta_x$, which is an element in $\hom(F(x), G(x))$ i.e. a morphism $F(x) \rightarrow G(x)$.

In your case the first functor is the identity, the second is the composition $FG$ and the functor $H$ is your $F$.