I've read about free cocompletion of categories discussing on the adjunction between Cat and cocompleteCat (Cat: category of small categories, cocompleteCat: category of small cocomplete categories and cocontinuous functors) where the adjunction is about a free cocompletion functor F which sends each category C to PSh(C), the category of its presheaves.
(<http://ncatlab.org/nlab/show/free+cocompletion>)


I wonder if the functor F have a left adjoint (even with some restrictions on its domain) and how it could be described.



a (pseudo)adjunction for the functor sending a category C to PSh(C) the category of presheaves
The free cocompletion functor $C \mapsto \widehat{C}$ (which, as Zhen Lin says, does not agree with the presheaf functor when $C$ is not essentially small) should in no reasonable sense have a left adjoint, since it is very far from preserving limits. 


It already fails to preserve products: if $C, D$ are two small categories, then $\widehat{C \times D}$ is the "cocomplete tensor product" of $\widehat{C}$ and $\widehat{D}$, which is neither the product nor the coproduct in cocomplete categories. (This is closely analogous to the way in which the free abelian group functor sends a product to a tensor product, which is neither the product nor the coproduct in abelian groups.) In fact the product of $\widehat{C}$ and $\widehat{D}$ is $\widehat{C \sqcup D}$. So we already get a counterexample by taking $C$ and $D$ to be discrete categories with $3$ objects, say. 


