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.

  • $\begingroup$ Do you want to exchange the roles of $\epsilon$ and $\eta$, or of $F$ and $G$? $\endgroup$ – Berci Sep 2 '19 at 17:51
  • $\begingroup$ So want to replace $\eta$ by something else to obtain an adjunction ? $\endgroup$ – Arnaud D. Sep 2 '19 at 19:21
  • $\begingroup$ @Berci The roles of $\epsilon$ and $\eta$. $\endgroup$ – Jxt921 Sep 2 '19 at 20:17
  • $\begingroup$ @ArnaudD. Exactly. $\endgroup$ – 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 $$\matrix{FGFx &\overset{F \eta GF}\to& FGFGFx \\ {\scriptstyle \varepsilon F}\downarrow \phantom{ {}_{\varepsilon F}} && \phantom{{}_{FG\varepsilon F}} \downarrow {\scriptstyle FG\varepsilon F}\\ Fx &\underset{F\eta}\to& FGFx}$$


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