TITLE: is localization of category of categories equivalent to |Cat|
QUESTION [9 upvotes]: It might be a stupid question. 
Suppose There is a category of categories,denoted by CAT,where objects are categories, morpshims are functors between categories
Take multiplicative system S={category equivalences}. Then we take localization at S. Then we get localized category S^(-1)CAT 
Another Category, denoted by |CAT|,where objects are categories, morphism are isomorphism classes of functors between categories. 
I want to prove this two categories are equivalent. I want to prove this problem in two ways:

Naive prove
Because either |CAT| or S^(-1)CAT has same objects as CAT. So one can prove the morphism class of these two category has same equivalent relationship. For the S^(-1)Cat, suppose F and G are two functors in the same equivalent class, we have sF=sG,for s belongs to S. Then I can prove F and G belongs to the same equivalent class in |CAT|. But on the other hand, if F and G are isomorphic functor. I can not prove sF=sG,for some s belongs to S. What I can only prove is sF is isomorphic to sG.
Using adjoint functors
Because I want to prove the projection functor CAT--->|CAT| is a localization functor at S. So I want to construct an adjoint functor |CAT|--->CAT which is fully faithful. But what is this functor. It seems I can not make it well defined.

This is an exercise in Toen's lecture: Lectures on DG-categories
I attach the lecture notes on DG-categories here Toen notes
page 5,exercise 2

REPLY [5 votes]: Bertrand Toën is using a different definition of localization than what you describe in your question. Let $\mathcal{C}$ be a category and let $\Sigma$ be a class of morphisms in $\mathcal{C}$. A category $\mathcal{D}$ with a functor $L\colon \mathcal{C} \rightarrow \mathcal{D}$ is called a localization of $\mathcal{C}$ at $\Sigma$ if $L(f)$ is an isomorphism for every $f \in \Sigma$, and $L$ is universal with this property, that is, whenever $F \colon \mathcal{C} \rightarrow \mathcal{E}$ is a functor such that $F(f)$ is an isomorphism for all $f\in \Sigma$, then there exists a unique functor $G \colon \mathcal{C} \rightarrow \mathcal{E}$ with $F=G \circ L$. If a localization of $\mathcal{C}$ at $\Sigma$ exists, then it is unique up to isomorphism and ususally denoted by $\Sigma^{-1}\mathcal{C}$ or $\mathcal{C}[\Sigma^{-1}]$.
Note that we do not make any assumptions about the class $\Sigma$. As with all universal properties, such an object need not exist in the most general case: usually the resulting category might be "too big", that is, there would be a proper class of morphisms between certain objects in the localization. If the class $\Sigma$ admits a right calculus of fractions in the sense of (e.g.) Borceux, Handbook of Categorical Algebra, Volume I, Chapter 5, then the localization of $\mathcal{C}$ at $\Sigma$ does exist. However, there are lots of examples of categories with classes $\Sigma$ which do have a localization but don't admit a right calculus of fractions. In many of these cases, the functor $L \colon \mathcal{C} \rightarrow \mathcal{D}$ will then not have a right adjoint.
One of the main sources of such categories are Quillen model categories $\mathcal{M}$, with $\Sigma$ the class of weak equivalences. The localization at $\Sigma$ is then called the homotopy category of $\mathcal{M}$. In fact, the notion of Quillen categories was invented mainly to deal with the difficulties of formally inverting a class of arrows in a category. There is a model structure on the category of small categories (sometimes called the folk model structure) whose weak equivalences are the equivalences of categories, so the desired localization does exist. Moreover, there is a notion of left and right homotopy in a model category. For nice objects (objects which are both fibrant and cofibrant), these notions agree and define an equivalence relation on maps. In the model category of categories, all objects are fibrant and cofibrant. Two functors are homotopic if and only if they are naturally isomorphic.
The usual construction of the homotopy category of a model category (see e.g. Mark Hovey's book "Model Categories") shows that the localization of $\mathbf{Cat}$ at the equivalences is given by the category whose objects are again categories and whose morphisms are homotopy classes of morphisms. Thus we do indeed find that the localization of $\mathbf{Cat}$ at the class of equivalences is the category $\vert \mathbf{Cat} \vert$ of categories and isomorphism classes of functors, as Toën claimed.
Charles Rezk has a nice writeup of the folk model structure here.