Usually when $F: C \rightarrow D$ is a localization functor, the categories $C$ and $D$ are not equivalent. My question is when is it possible for $C, D$ to be abstractly equivalent but $F$ is not an equivalence. Alternatively, when is there a localization functor $F: C \rightarrow C$ that is an endofunctor but not an equivalence?
I think such examples can exist when $C$ is of ``infinite type". For example, if $C \cong \oplus_{\mathbb{N}} D$, then there is a non-trivial localization $C \rightarrow C$ obtained by making the first copy of $D$ trivial. I would guess that if $C$ has some compactness condition, then any localization $F: C \rightarrow C$ must be an equivalence.
More precisely, I am interested in the examples when $C$ is a presentable stable $\infty$-category so that $F$ is equivalent to making some objects $A$ of $C$ trivial, i.e. $C \cong C/A$, and $C$ is smooth (which is a compactness condition).
