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Begin with an equivalence $F: C \to D$, $G: D \to C$ along with natural isomorphisms_ to the identities, so $\eta: 1_C \simeq GF$ and $\epsilon: FG \simeq 1_D$. The claim is that we can replace $\epsilon$ by $\epsilon'$ such that the resulting ($\eta$, $\epsilon'$) pair obey the triangle identities.

Begin by defining $\gamma \equiv G \overset{\eta G}{\rightarrow} GFG \overset {G \epsilon}{\rightarrow} G$. This need not be identity, and will not be if $(\eta, \epsilon)$ is not an adjunction. So define $\epsilon' \equiv \epsilon \circ F\gamma^{-1}$.

The claim now is that the map $\epsilon'_F \circ F \eta$ is idempotent.

  1. This appears like a typo to me. Should it not be $\epsilon' F \circ F \eta$ is idempotent, from the triangle identity? (notice that the $F$ is not a subscript of $\epsilon'$).

Next, the claim is that an idempotent isomorphism is an identity.

  1. Does this follow because $s^2 = s$ where $s$ is an iso implies that $s^2 \circ s^{-1} = s \circ s^{-1}$, or $s = id$?

Finally, the claim is that $\epsilon' F \circ F \eta$ is idempotent follows by chasing the diagram:

enter image description here

  1. What is $\eta_{GF}, \epsilon'_F, \epsilon'_{FGF}$? I imagine that it really means $\eta GF$, $\epsilon' F$, and $\epsilon' FGF$?

  2. How does one conveniently show that the diagram really does commute?

  3. And finally, I imagine that there must be an easier way to prove these theorems without chasing commutative diagrams? Do string diagrams help? Some other formalism?

Snippet, Proposition 4.4.5, Category theory in Context

enter image description here

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(1) and (3): For a natural transformation $\alpha$ and a (composable) functor $T$ we have by definition $$(\alpha T)_x:=\alpha_{T(x)}$$ this explains why the notations $\alpha T$ and $\alpha_T$ mean the same thing.

(2): Yes, correct.

(4): Each square commutes because of a naturality condition of one of the natural transformations. And the lower triangle commutes because of definition of $\epsilon'$.

(5): Well, yes, one can do a proof using string diagrams or general bicategorical arguments, though I wouldn't say they are 'simpler' or 'easier' (at least to me).

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  • $\begingroup$ Could you elaborate how the lower triangle commutes because of the definition of $\epsilon'$? I don't quite follow. $\endgroup$ Nov 9, 2021 at 0:25

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