Size issue in exhibiting the free cocompletion as a left adjoint Consider the following well-known statement:

Let $C$ be a small category, $E$ a cocomplete category, and $F\colon C\to E$ a functor. Then there is a (up to isomorphism) unique cocontinuous functor $F'\colon \mathbf{Set}^{C^\mathrm{op}}\to E$ such that $F'\circ y \cong F$, where $y\colon C\to \mathbf{Set}^{C^\mathrm{op}}$ is the Yoneda embedding.

Question: Is there a way to phrase this as an adjunction (between bicategories, of course)?
Idea: Consider the bicategory $\mathbf{SmallCat}$ of small categories and functors and the bicategory $\mathbf{CocompCat}$ of cocomplete categories and cocontinuous functors. Then the above statement almost states that $C\mapsto \mathbf{Set}^{C^\mathrm{op}}$ is a left adjoint to the forgetful functor $\mathbf{CocompCat}\to \mathbf{SmallCat}$, except that this forgetful functor isn't well-defined: the underlying category of a cocomplete category isn't small.
How can one fix this size issue?
 A: As Denis-Charles says in the comments, the best way to handle this is to replace the presheaf category $\mathbf{Set}^{C^{\mathrm{op}}}$ by the full subcategory $\hat{C}$ of small presheaves. By definition, a presheaf is small if it satisfies any of these equivalent conditions:

*

*it is a small colimit of representables;


*it is the left Kan extension of its restriction to some small full subcategory of $C$;


*it is the left Kan extension of some presheaf on some small category along some functor into $C$.
Every presheaf on a small category is small. But, for instance, a presheaf on a large discrete category is small iff its support is small; hence the terminal presheaf is not small.
The functor $C \mapsto \hat{C}$ is left adjoint to the forgetful functor
$$
(\text{cocomplete locally small categories}) \to (\text{locally small categories}),
$$
in a suitable 2-categorical sense.
A standard reference for this is:

Brian J. Day and Stephen Lack. Limits of small functors. Journal of Pure and Applied Algebra 210 (2007), 651–663.

But it goes back further than 2007. The introduction to Day and Lack's paper recounts some of the history.
