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

rightadjoint, the forgetful functor from cocomplete categories to arbitrary categories, are discussed. The size issues which Qiaochu refers to are ignored though. $\endgroup$ – Ingo Blechschmidt May 2 '15 at 21:33largecategories with small colimits to large categories has a genuine left adjoint which sends a small category C to PSh(C). But this left adjoint is not a right adjoint: PSh(*) is not final, since it has many colimit-preserving endofunctors. $\endgroup$ – Marc Hoyois May 2 '15 at 22:02