Let $\mathcal{U}$ be a universe and $\mathcal{U}^+$ a universe with $\mathcal{U} \in \mathcal{U}^+$. Denote by $\text{Cat}(\mathcal{U})$ the $\mathcal{U}^+$-category of all $\mathcal{U}$-categories, and by $\text{Cat}_c(\mathcal{U})$ the full subcategory consisting of $\mathcal{U}$-cocomplete categories, i.e. in which $\mathcal{U}$-small colimits exist. Consider $\mathcal{U}$ as the $\mathcal{U}$-category of all sets in $\mathcal{U}$.

**Question.** Does the inclusion $\text{Cat}_c(\mathcal{U}) \to \text{Cat}(\mathcal{U})$ have a left adjoint?

Remark that if $C \in \text{Cat}(\mathcal{U})$, then $\widehat{C} = \text{Hom}(C^{\text{op}},\mathcal{U})$, what is usually called the universal cocompletion of $C$, is a $\mathcal{U}^+$-category (the universum jumps!) which is $\mathcal{U}$-cocomplete, which satisfies the following universal property: $\mathcal{U}$-Functors $C \to D$, where $D$ is a $\mathcal{U}$-cocomplete $\mathcal{U}^+$-category, correspond to $\mathcal{U}$-cocontinuous $\mathcal{U}^+$-functors $\widehat{C} \to D$. Thus we have no adjointness property in the usual sense, right?

Actually I doubt that there is a left adjoint in the usual sense. But often universal cocompletions are quite useful and seem to be used without any discussion of the set-theoretic problems above (for example one restricts to small categories). Is that because it's not bad that we leave the universe, at least in some contexts?

Or is it possible to repair this? Is there a universe $\mathcal{U}$ which is big enough so that the above jumps "stabilize" below $\mathcal{U}$? Perhaps we can use Mahlo cardinals? Or can we repair this by restricting to finite colimits? Isn't this more reasonable since the definition of a universe is finitary? How does the left adjoint to $\text{Cat}_{fc}(\mathcal{U}) \to \text{Cat}(\mathcal{U})$ look like explictly, if it exists?

So my question is basically about the set theoretic subtletlies behind the universal cocompletion of a category. Feel free to write anything you know about them ...