Is there a standard term for a morphism of sites $(C,J)\to(C',J')$ which induces an equivalence on sheaf categories $\operatorname{Sh}(C',J')\xrightarrow\sim\operatorname{Sh}(C,J)$? The [nLab entry](https://ncatlab.org/nlab/show/site) suggests the term **Morita equivalence**, however I have not yet found any other sources which use this term in this way.



What to call a morphism of sites inducing an equivalence on categories of sheaves?
Johnstone's *Sketches of an Elephant* (2 volumes) is a standard reference which uses "Morita equivalence" in this way. In fact, Jonstone systematically uses "Morita equivalence" in a similar way across many types of categorical logic. Relevant here is the case of *geometric logic*.


Although some of the terminology in the *Elephant* is a bit idiosyncratic, I think this use of "Morita equivalence" is pretty common in the categorical logic literature. And the analogy to ring theory which it brings to mind is spot-on, I think.


