I'm studying category theory by myself and I just came across this sentence from [Wikipedia](https://en.wikipedia.org/wiki/Adjoint_functors):


*An adjunction between categories C and D is somewhat akin to a "weak form" of an equivalence between C and D, and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.*


Can someone provide an example of an "upgrade" of an adjunction to an equivalence? I'm interested in understanding why I could intuitively think of an adjunction as a weak form of an equivalence.



Upgrade adjunction to equivalence
Let $\mathcal C$ and $\mathcal D$ be two categories, and let $F\colon\mathcal C\longrightarrow \mathcal D$ and $G\colon\mathcal D\longrightarrow\mathcal C$ be two functors, with $F$ left adjoint to $G$. Then there are natural transformations $F\circ G\longrightarrow \operatorname{Id}\_{\mathcal D}$ and $\operatorname{Id}\_{\mathcal C}\longrightarrow G\circ F$ (as mentioned in a comment above).


Let $\mathcal A\subset\mathcal C$ be the full subcategory consisting of all objects $A\in\mathcal C$ for which the natural morphism $A\longrightarrow GF(A)$ is an isomorphism. Similarly, let $\mathcal B\subset\mathcal D$ be the full subcategory consisting of all objects $B\in\mathcal D$ for which the natural morphism $FG(B)\longrightarrow B$ is an isomorphism.


Then one can check that $F(\mathcal A)\subset\mathcal B$ and $G(\mathcal B)\subset\mathcal A$. The restrictions of the adjoint functors $F$ and $G$ to the full subcategories $\mathcal A\subset\mathcal C$ and $\mathcal B\subset\mathcal D$ are again adjoint functors: the functor $F|\_{\mathcal A}\colon \mathcal A\longrightarrow\mathcal B$ is left adjoint to the functor $G|\_{\mathcal B}\colon \mathcal B\longrightarrow\mathcal A$. The adjunction between the functors $F|\_{\mathcal A}$ and $G|\_{\mathcal B}$ is an equivalence between the categories $\mathcal A$ and $\mathcal B$,
$$
 F|\_{\mathcal A}\colon\mathcal A\,\simeq\,\mathcal B:\!G|\_{\mathcal B}.
$$
This result can be found in the paper A. Frankild, P. Jorgensen, "Foxby equivalence, complete modules, and torsion modules", J. Pure Appl. Algebra 174 #2, p.135-147, 2002, <https://doi.org/10.1016/S0022-4049(02)00043-9> , Theorem 1.1.


I am not sure whether this should be properly called "upgrading an adjuction to an equivalence", though. The passage from the adjoint pair $(F,G)$ to the equivalence $(F|\_{\mathcal A},\,G|\_{\mathcal B})$ entails losing rather than gaining information. Perhaps it would be better to call it "restricting an adjunction to an equivalence".


Then again, I do not know what the author of the passage in the Wikipedia article might have had in mind.


