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Given two categories $C$ and $D$ and two fucntors $F:C \to D$ and $G:D \to C$ that form an equivalence. Is it true that there is more than one way to describe the equivalence in terms of a unit and a counit? If so then what can one say about the relationship between the these different units and counits? Is there some type of classifying space?

Dick Johnson
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  • You mean $F$ and $G$ form an adjunction? – Jeroen van der Meer Jan 27 '21 at 13:22
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    Take an easy example: consider the case where both categories are groups considered one-object groupoids. An equivalence of categories here is basically a pair of isomorphisms of groups whose composites are inner automorphisms, and the choice of units is basically the kernel of $G \to \operatorname{Inn} G$. – Zhen Lin Jan 27 '21 at 13:29
  • I have written some details in the related question of yours: https://math.stackexchange.com/questions/3891297 – Martin Brandenburg Apr 24 '21 at 18:20

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