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I'm reading a book on category theory by Tom Leinster and I've encountered something a little weird. It states that when $(F, G, \eta, \epsilon)$ is an equivalence, it is true that $F$ is left adjoint to $G$, but that the unit and counit are not necessarily $\eta$ and $\epsilon$.

Later on, it shows a proof that there's a one to one correlation between natural transformations for which the element at $A$ is an initial object in the comma category $(A \Rightarrow G)$ and adjunctions. From what I've understood such natural transformation would also be the unit of the adjunction.

Later on, I've proved that for an equivalence if $(F, G, \eta, \epsilon)$, $(A, \eta_A)$ is an initial object in $(A\Rightarrow G)$ for all $A$ and from what I understand it also makes it the unit of an adjunction.

But I've seen that $\eta$ and $\epsilon$ might not be the unit and counit of an adjunction? Does this mean that there are equivalences where if I take, $\eta$ to be the unit, then $\epsilon$ can't be the counit? Or am I wrong in my assumptions?

Sorry if this question is a bit silly I'm a beginner with all this, still trying to wrap my head around it.

Philippe 7433
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  • Yes, $\eta$ is a unit for an adjoint $(\eta,\epsilon')$ of $(F,G)$, for some unique natural isomorphism $\epsilon'\colon FG\to\operatorname{id}$, which may not equal $\epsilon$. See this answer(https://math.stackexchange.com/a/595487/1173024) which links to a proof. – Noiril Apr 11 '25 at 23:54
  • See also adjoint equivalences. – Naïm Camille Favier Apr 12 '25 at 01:18
  • Thank you both for your comments, if you want to post it as an answer I'll mark it as answering my post – Philippe 7433 Apr 12 '25 at 03:16
  • Indeed if $(F,G,\eta,\epsilon)$ is equivalence then $F$ is left adjoint to $G$ with unit $\eta$, but counit of this adjunction may be different from $\epsilon$. You can reverify your proof with my answer. You can similarly write equivalent definition of adjointness as natural transformation $\epsilon$ such that $\epsilon_B$ is initial object. With this reformulation, $F$ is left adjoint to $G$ with counit $\epsilon$, but unit of this adjunction may be different from $\eta$. – user264745 May 03 '25 at 22:29

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