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Oh, Toën is using a different definition of "localization" than the one I am accustomed to (e.g., see Definition 5.2.7.2 and Warning 5.2.7.3 of Higher Topos Theory). Since Cat is a presentable category, I assumed you were taking the localization in presentable categories, which involves a category S-1Cat and an adjunction with left adjoint Cat S-1Cat which has the analogous universal property with respect to adjunctions between presentable categories. In that case, S-1Cat turns out to be rigid categories and Chris explained why you might expect this to be true. (The homotopy category of Cat is presumably not a presentable category at all.)

With S the class of equivalences of categories these localizations do not agree because S is not closed under pushouts in Cat: in the square

J -> BZ* -> *the left-hand map is in S but the right-hand map is not and yet the square is, bizarrely, a pushout.  (What makes things even trickier is the ∞-categorical analogue of "S is closed under pushouts" is true because of course we must take homotopy pushouts.  In short thinking of Cat as a 1-category and trying to work with equivalences there can lead to strange results!)

 
 
 
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I'm pretty sure this is false. If I recall correctly, the localization S-1CAT you refer to is equivalent to the category of rigid categories, those with no nonidentity isomorphisms at all. In particular, the "rigid categorification" of any connected groupoid is the terminal category. On the other hand, the isomorphism classes of objects in |CAT| are equivalence classes of categories, so in |CAT| there are many nonisomorphic connected groupoids.

I'll see if I can remember how to prove my claim about S-1CAT, or maybe you can prove it; I don't think it's very hard, and it uses the same kind of ideas Chris talked about in his answer.