# Promoting equivalence to adjoint equivalence: explicit construction of a unit and a counit

Let $$F\colon\mathsf{C}\to\mathsf{D}, G\colon\mathsf{D}\to\mathsf{C}$$ together with $$\eta\colon 1_{\mathsf{C}}\to \mathsf{GF}, \epsilon\colon\mathsf{FG}\to 1_{\mathsf{D}}$$ be an equivalence of categories. I know how to replace $$\epsilon$$ with $$\epsilon'$$ and such $$\epsilon'$$ is still an natural isomorphism and together with $$\eta$$ they form a unit-counit pair.

Howerver, for some reason I can't figure out with what we should replace $$\eta$$ if we want that instead. Riehl in the book Category Theory in Context defines $$\gamma = G(\epsilon)\circ \eta G$$ and claims that

Redefining either $$\epsilon$$ or $$\eta$$ so as to absorb the isomorphism $$\gamma$$ - it will not matter which -we will show that there sulting pair of natural isomorphisms define the unit and the counit of an adjunction $$F\dashv G$$.

Then she replaces $$\epsilon$$ with $$\epsilon' = \epsilon\circ F(\gamma^{-1})$$ and proves that $$\eta$$ and $$\epsilon'$$ satisfy the triangle identites for the unit and the counit. I tried replacing $$\gamma$$ with $$\epsilon F\circ F(\eta)$$ to no avail.

• Do you want to exchange the roles of $\epsilon$ and $\eta$, or of $F$ and $G$? – Berci Sep 2 '19 at 17:51
• So want to replace $\eta$ by something else to obtain an adjunction ? – Arnaud D. Sep 2 '19 at 19:21
• @Berci The roles of $\epsilon$ and $\eta$. – Jxt921 Sep 2 '19 at 20:17
• @ArnaudD. Exactly. – Jxt921 Sep 2 '19 at 20:17

You don't need to replace $$\gamma$$.
Set $$\ \eta':=\gamma^{-1}F\circ\eta$$.
Then we have $$G(\varepsilon)\circ \eta'G\ =\ G(\varepsilon)\circ \eta^{-1}GFG \circ G\varepsilon^{-1}FG\circ\eta G$$. $$\matrix{GFGx &\overset{\eta GFG}\to& GFGFGx \\ {\scriptstyle G\varepsilon}\downarrow \phantom{ {}_{G\varepsilon}} && \phantom{{}_{G\varepsilon FG}} \downarrow {\scriptstyle G\varepsilon FG}\\ Gx &\underset{\eta G}\to& GFGx}$$ Show that this square is commutative, and as each arrow is invertible, the above composition is $$1_{Gx}$$.
Similarly, $$\varepsilon F\circ F(\eta')\ =\ \varepsilon F\circ F\eta^{-1}GF\circ FG\varepsilon^{-1}F\circ F\eta\ =\ 1_{Fx}$$, belonging to the commutativity of $$\matrix{FGFx &\overset{F \eta GF}\to& FGFGFx \\ {\scriptstyle \varepsilon F}\downarrow \phantom{ {}_{\varepsilon F}} && \phantom{{}_{FG\varepsilon F}} \downarrow {\scriptstyle FG\varepsilon F}\\ Fx &\underset{F\eta}\to& FGFx}$$