Given a class of arrows in a category, the localization (if it exists) inverts all the arrows in , i.e. is an isomorphism in and is universal with respect to all functors with such a property. The universality makes it a 2-categorical colimit called a coinverter in the meta-2-category of large 2-categories. The universality says that every -inverting functor factors through up to an iso 2-cell; i.e. for every which is inverting all arrows in , there exist a functor such that ; in addition, for all categories , the functor of precomposing with is fully faithful functor from (it follows that is unique up to unique iso). It follows that the image of is precisely the subcategory of consisting of -inverting functors.
We say that is a strict localization (functor) if for all , the corresponding can be chosen in a unique way such that . It follows that every two strict localizations are not only equivalent but in fact isomorphic. For example, when admits the calculus of (say, left) fractions then both the large construction via the path category and the construction via one sided fractions give the isomorphic categories. The additional requirement that for each , is fully faithful automatically follows essentially from the uniqueness of satisfying strict equality (or, alternatively, for small categories, using lemma 1.2 in Gabriel-Zisman which claims this for the construction of the localized category via the path category, and the fact that every two strict localizations are isomorphic).