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If $\mathcal{C}$ is a category, there is a well known construction called the Ind-completion of $\mathcal{C}$, indicated by $\text{Ind}(\mathcal{C})$. This can be equivalently defined in several ways: for example, $\text{Ind}(\mathcal{C})$ may be seen as the full subcategory of $[\mathcal{C}^{op}, \mathbf{Set}]$ whose objects are those presheaves which are isomorphic to filtered colimits of representables. This construction has a universal property: namely, each functor $\mathcal{C}\rightarrow\mathcal{D}$, with $\mathcal{D}$ a category with filtered colimits, factors uniquely through the inclusion $\mathcal{C}\hookrightarrow\text{Ind}(\mathcal{C})$.

Now let $\mathcal{C}$ be a 2-category, and consider the 2-category of pseudofunctors $F:\mathcal{C}^{op}\rightarrow\mathbf{Cat}$ which are isomorphic to filtered pseudocolimits of representables, with the usual pseudonatural transformations and modifications.

My question is: has this 2-category ever been studied? Does it have a universal property similar to the one of the Ind-completion of a category?

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You can find discussion of this in Lack's 2-categories companion. The original reference in a more general context is Kelly's Structures defined by finite limits in the enriched context. See more discussion in Lack and Rosicky's Notions of Lawvere Theory.

The theory is a bit more delicate because often one really wants to work up to 2-equivalence, but the most "automatic" way to develop the theory views 2-categories as categories enriched in $Cat$, so the natural notion of equivalence is stricter.

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