# Category theory in Context, Proposition 4.4.5: Idemotence when Promoting equivalence to adjunction

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: 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 (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.
(4): Each square commutes because of a naturality condition of one of the natural transformations. And the lower triangle commutes because of definition of $$\epsilon'$$.
• Could you elaborate how the lower triangle commutes because of the definition of $\epsilon'$? I don't quite follow. Nov 9, 2021 at 0:25