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This is Exercise 6.14 from A Term of Commutative Algebra by Altman and Kleiman.

Let $\mathcal{C}$ and $\mathcal{C}'$ be categories, $F:\mathcal{C}\to\mathcal{C}'$ and $F':\mathcal{C}'\to\mathcal{C}$ be an adjoint pair. Let $\varphi_{A,A'}:\text{Hom}_{\mathcal{C}'}(FA,A)\to\text{Hom}_{\mathcal{C}}(A,F'A')$ denote the natural bijection, and set $\eta_A:=\varphi_{A,FA}(1_{FA})$. Do the following:

(1) Prove $\eta_A$ is natural in $A$; that is, given $g:A\to B$, the induced square $\require{AMScd}$ \begin{CD} A @>\eta_A>> F'FA\\ @V g VV @VV F'Fg V\\ B @>>\eta_B> F'FB \end{CD} is commutative. We call the natural transformation $A\mapsto\eta_A$ the unit of $(F,F')$.

(2) Given $f':FA\to A'$, prove $\varphi_{A,FA}(f')=F'f'\circ\eta_A$.

(3) Prove the canonical map $\eta_A:A\to F'FA$ is universal from $A$ to $F'$; that is, given $f:A\to F'A'$ there is a unique map $f':FA\to A'$ with $F'f'\circ\eta_A = f$.

(4) Conversely, instead of assuming $(F,F')$ is an adjoint pair, assume given a natural transformation $\eta:1_\mathcal{C}\to F'F$ satisfying (1) and (3). Prove the equation in (2) defines a natural bijection making $(F,F')$ an adjoint pair, whose unit is $\eta$.

My confusion is on (4). I understand how the equation in (2) defines a bijection using (3), but I don't understand why this is a natural bijection. The solution in the back defines the bijection $\psi_{A,A'}:\text{Hom}_{\mathcal{C}'}(FA,A')\to\text{Hom}_{\mathcal{C}}(A,F'A')$ and then goes on to say "Also, $\psi_{A,A'}$ is natural in $A$, as $\eta_A$ is natural in $A$ and $F'$ is a functor. And, $\psi_{A,A'}$ is natural in $A'$, as $F'$ is a functor. Clearly, $\psi_{A,FA}(1_{FA})=\eta_A$."

My question can be summed up as (i) how is $\psi_{A,A'}$ a natural bijection (which I assume to mean a natural transformation that is bijective) and (ii) what is meant by $\psi_{A,A'}$ is natural in $A$ and $A'$?

Any clarification that can be provided on this would be very much appreciated, I suspect I'm missing something relatively straightforward. I am self studying this material and not currently in school so I don't have anyone I can easily ask. Thank you!

iceberg56
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  • Usually people use "natural" as an adjective if either (1) the construction made no choices or (2) if there is commutative property. So in this case, if there was a map $A\to B$, then there is an appropriate diagram with $\psi_{A,A'}$ and $\psi_{B,A'}$ appearing that commutes. Same thing if there is a map $A'\to B'$. Right now I'm too lazy to write anything more detailed so in the mean time, maybe check out related posts such as this one https://math.stackexchange.com/questions/835870/proving-that-the-transformation-obtained-from-an-adjoint-pair-is-natural?rq=1 –  Nov 23 '23 at 02:01

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