Consider the following well-known statement:



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> 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.
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*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?



Size issue in exhibiting the free cocompletion as a left adjoint
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:



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> Brian J. Day and Stephen Lack. [Limits of small functors.](https://arxiv.org/abs/math/0610439) *Journal of Pure and Applied Algebra* 210 (2007), 651–663.
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But it goes back further than 2007. The introduction to Day and Lack's paper recounts some of the history.


