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 ...



Universal cocompletion without leaving our universe
you can generalizing the $Ind(\mathcal{C})$ construction (see SGA.4-I) for general small diagrams (no neccessarly directed), using the general criterion that characterizes the purpose of a diagram $ D: I \to \mathcal{C} $ in terms of connecting category paragraph 
$ X \downarrow D, \ X \in \mathcal{C} $). In this way you get a cocompletion of a (big) category $\mathcal{C}$ relatively to the small colimits (without escape from base universe).


Is this generalization $Ind(\mathcal{C})$ is equivalent to the category $\mathcal{C}^>$ of presheaves on $\mathcal{C}$ formed by the elements $P\in \mathcal{C}^>$ such that the comma category (said also the Grothendieck costruction) $\mathcal{C}\downarrow P$ ha a small final subcategory.


You can find this construction in:


Shape Theory: Categorical Methods of Approximation di J. M. Cordier, T. Porter.


Or if you can read Italian language, I send you a my script about Category theory (involving aspect of completions of categories).\


This a Sketch:


Then give a morphism $\underline{f}: P \to Q$ we can represent it chosing for any $i\in I$ a lifting $h\_{X\_i} \to h\_{Y\_j(i)}$ i.e. $X\_i \to Y\_j$ of the morphism $h\_{X\_i} \to P \to Q$ then we have a map $\phi : |I|\_0 \to |J|\_0$ and a family of morphism $f\_i: X\_i \to Y\_{\phi (i)}$ such that for a arrow $i \to i'$ in $I$ the morphisms $f\_i: X\_i \to Y\_{\phi(i)}$ and $X\_i \to X\_{i'} \to Y\_{\phi(i') }$ are connected as objects of $X\_i \downarrow \underline{Y}$, and a such data make a unique (but Iso) morphism $P \to Q$.
We call $(\phi , (f\_i)\_i) $ a representation of $\underline{f}$, and two representation $(\phi , (f\_i)\_i), \ (\phi' , (f'\_i)\_i) $ of $\underline{f}$ correspond to the some morphism $P \to Q $ IFF for every $i\in I$ the morphisms $ f\_i: X\_i \to Y\_{\phi(i) },\ f'\_i: X\_i \to Y\_{\phi'(i) } $ are connected in
 $X\_i \downarrow \underline{Y}$.


To a composition $\underline{g}\circ \underline{f}: P \to Q \to R$ we have in terms of representation:


$(\psi, (g\_j))\circ(\phi,(f\_i)) = (\psi\circ \phi,(g\_{\eta(i)}\circ f\_i)\_i)$


Then we have the category of representations data as above, and call it $Ind\_0(\mathcal{C})$ the (generalized Ind-category os $\mathcal{C}$, and a equivalence $Ind\_0(\mathcal{C}) \sim P\_0(\mathcal{C})$.


Now is $\mathcal{A}$ has small colimits any funtors $F: \mathcal{C} \to \mathcal{A}$ has a unique (but ISo) extention to a colimits preserving funtors: $P\_0(F): \mathcal{C} \to \mathcal{A}$ where $P\_0(-)$ is the (puntual) left-Kan-extention , this is "computable" in terms of Ind-representation: let give $\underline{f}: P \to Q$ in $P\_0(\mathcal{C})$ as above, then we can define $P\_0(P)= {\underrightarrow{lim}}\_{i\in I} F(X\_i) $, $P\_0(Q)= {\underrightarrow{lim}}\_{j\in J} F(Y\_j) $, and $P\_0(\underline{f})$ inducted by the family $F(f\_i): F(X\_i)\to F(Y\_{\phi (i)}$ (follow a well defined morphism $P\_0(\underline{f})$ for the connection relation described above).


Excuse my poor English.. 


