a (pseudo)adjunction for the functor sending a category C to PSh(C) the category of presheaves 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: 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. 
A: Here is the correct statement: 

Let $\mathfrak{Cat}$ be the 2-category of locally small categories (and all functors) and let $\mathfrak{Cocomp}$ be the 2-category of locally small cocomplete categories (and all cocontinuous functors). Then the forgetful functor $\mathfrak{Cocomp} \to \mathfrak{Cat}$ has a left biadjoint. More precisely, for every locally small category $\mathcal{C}$, there is a locally small cocomplete category $\hat{\mathcal{C}}$ and a functor $\mathcal{C} \to \hat{\mathcal{C}}$ such that the induced functor
  $$\mathfrak{Cocomp} (\hat{\mathcal{C}}, \mathcal{D}) \to \mathfrak{Cat} (\mathcal{C}, \mathcal{D})$$
  is an equivalence of categories for all locally small cocomplete $\mathcal{D}$.

In the case where $\mathcal{C}$ is essentially small, $\hat{\mathcal{C}}$ is equivalent to $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$; in general, it is a certain full subcategory.
(Note that we may take "locally small" in the weak sense of "hom-sets are small sets", without any condition on the set of objects.)
A: 
In the case where $C$ is essentially small, $\hat{C}$ is equivalent to $[C^\text{op}, \mathbf{Set}]$; in general, it is a certain full subcategory.

Let me make this statement precise, as it is actually the tricky part.
A presheaf $H\colon C^\text{op} \to Set$ is called small if $H = \mathrm{Lan}_JF$ for some full inclusion $J\colon D \hookrightarrow C^\text{op}$. Equivalently, a presheaf is a small colimit of representables (in a bigger universe). Roughly speaking, $H$ is determined by a small full subcategory of $C$. 
Then, the category $\mathcal{P}C$ of small presheaves with natural transformations forms the free cocompletion of $C$ via the usual Yoneda embedding. This gives you the left biadjoint to the forgetful functor $U\colon \mathbf{ConCAT} \to \mathbf{CAT}$ where $\mathbf{ConCAT}$ is the 2-category of large categories with colimits as objects, cocontinuous functors as 1-cells, and natural transformations as 2-cells; $\mathbf{CAT}$ the 2-category of large categories. In addition, the forgetful functor is also (pseudo-)monadic. The pseudomonad can be a strict 2-monad if a choice of colimits is given.
Small functors were introduced as accessible functors in (Kelly, 1982), and the monadicity of cocompletion can be found in (Kelly & Lack, 2000). 


*

*Kelly, Basic Concepts of Enriched Category Theory, 1982

*Kelly and Lack, On the monadicity of categories with chosen colimits, 2000

