For the purposes of this question, let me fix a "true" universe of sets, which I will call the "true sets". Recall that a category is locally presentable if it is cocomplete and accessible. Both these notions involve sets --- I will interpret them in terms of the true sets. Indeed, the word "category" involves sets (there should be a set of morphisms between any two objects) and again here I mean true sets.
Now choose some other model of set theory, which I will call the "fake sets". I presume that the fake sets assemble into a category, at least if the fake model is built "internally" to the true model.
Under what circumstances is the category of fake sets locally presentable?