I agree that [Leonid Positselski’s first answer](https://mathoverflow.net/a/316665/2273) seems probably what the writer had in mind: given an adjunction, restricting to the categories of “fixed points” on each side yields an equivalence.  Here are two important examples in nature, both involving the category of topological spaces:

- There’s an adjunction between the categories of preordered sets and topological spaces, sending a preordered set $(X,\leq)$ to $X$ with the topology of down-closed sets, and sending a topological space $Y$ to $Y$ with its specialisation order.  All preorders are fixpoints; on the other side, the fixpoints are exactly the *Alexandrov* spaces, i.e. spaces where arbitrary intersections of opens are open.  Restricting to this subcategory shows that the category of preorders is equivalent (in fact, isomorphic!) to the category of Alexandrov spaces.

- The adjunction between the categories of topological spaces and *locales*, sending a topological space to its frame/locale of opens and sending a locale to its space of points, restricts to the equivalence between *spatial locales* and *sober spaces*.