Let us denote by $\text{Cat}_c$ the $2$-category, whose objects are cocomplete categories, morphisms are cocontinuous functors and morphisms are natural transformations. Is it then true that $\text{Cat}_c$ is $2$-cocomplete, i.e. has every $2$-functor $C : I \to \text{Cat}_c$, where $I$ is a small $1$-category, a $2$-colimit?

I have the following idea: Let $U :\text{Cat}_c \to \text{Cat}$ be the forgetful $2$-functor. Now $\text{Cat}$ is $2$-cocomplete; a comprehensive reference for this is "The stack of microlocal perverse sheaves" by Ingo Waschkies. Besides, $U$ has a $2$-left adjoint $\widehat{(- )} : \text{Cat} \to \text{Cat}_c$ (see here). So we may consider the cocomplete category $D:=\widehat{\text{colim} UC}$, but of course the functors $C_i \to D$ are not cocontinuous. Thus perhaps we have to define some reflective (and hence cocomplete) subcategory $D'$ of $D$, such that each $C_i \to D$ factors through a cocontinuous functor $C_i \to D'$. But I don't know how to define $D'$.

I'm also interested in colimits in similar $2$-categories, for example $\text{Cat}_{c\otimes}$, which consists of the cocomplete tensor categories. And actually I want something more, namely that these $2$-categories are $2$-locally presentable. Any hints and references are appreciated!

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