Clarifying the concept of the unit of an adjunction

adjoint-functorscategory-theory

$\newcommand{\Hom}{\operatorname{Hom}}\newcommand{\iso}{\texttt{iso}}\newcommand{\id}{\operatorname{id}}$
I'm reading Emily Riehl's "Category Theory in Context", Section 4.2, "The unit and counit as universal arrows". I transcribe my understading of her development of the unit-counit formulation of an adjunction, and then explain what I am confused about.

The development

We begin with the first definition of an adjunction, which is functors $F: C \to D$, $G: D \to C$ along with a natural isomorphism of Hom-sets $\Hom(Fc, d) \simeq \Hom(c, Gd)$ for all $c \in C, d \in D$.

We analyze the situation by allowing $d$ to vary freely. This gives us the natural transformation $\Hom(Fc, -) \simeq Hom(c, G(-))$. This implies that the functor $\Hom(c, G(-))$ is represented by the $\Hom$-functor $\Hom(Fc, -)$. Let's call the natural isomorphism as $\iso: \Hom(Fc, -) \simeq \Hom(c, G-)$. By Yoneda, we know that such a natural transformation $\iso$ is determined entirely by the image of $\id_{Fc}$ under $\iso$. Thus, we can determine $\iso$ by determining $\iso(\id_{Fc}) \in \Hom(c, G(Fc))$.

Next, we decide to "let $c$ vary in the above expression". We can do this by thinking of $\Hom(c, GFc)$ as $\Hom(1_C c, GFc)$, and then letting $c$ vary. This is asking for a natural transformation $\eta: 1_C \Rightarrow GF$. This $\eta$ is the "unit of the adjunction". We define $\eta$ pointwise as $\eta_c \equiv \iso(\id_{Fc}) \in \Hom(c, GFc)$.

I am quite perplexed by this leap of faith, where we introduce $\eta: 1_C \Rightarrow GF$ by generalizing the original representing object. $\iso(\id_{Fc})$. What motivates this generalization?

Questions

  • Did I get the story correctly, or did I miss some aspect of the definitions / the development of the idea of $\eta: 1_C \Rightarrow GF$?

  • Is it possible to explain why I should have thought of creating $\eta: 1_C \Rightarrow GF$ after observing that $\iso(id_{Fc})$ contains all the information of $\iso: \Hom(Fc, -) \simeq \Hom(c, G(-))$?

Screenshot of the text:

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Best Answer

This is good thinking so far. I don't think it's hard to finish your train of thought.

The introduction of $\eta_c$ shows you how to encapsulate the entire natural isomorphism $\mathrm{Hom}(Fc,-)\cong \mathrm{Hom}(c,G-)$ into a single morphism. This gives you isomorphisms $\mathrm{Hom}(Fc,d)\cong\mathrm{Hom}(c,Gd)$ for every $c$ and $d,$ natural in $d$. But that's not what an adjunction is. These isomorphisms are also natural in $c,$ so to recover this property of an adjunction we're forced to ask whether in some sense $\eta_c$ is itself natural in $c.$

Edit: Note that the sequencing of the presentation above obfuscates what exactly $\eta$ provides you with, which is merely a natural transformation $\mathrm{Hom}(F-,-)\to \mathrm{Hom}(-,G-).$

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