Let $\mathscr K$ be a small 2-category. It follows from $\mathrm{Cat}$-enriched category theory that the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors is given by the 2-category $[\mathscr K^\circ, \mathrm{Cat}]$ of 2-functors and 2-natural transformations.
In particular, strict 2-colimits subsume other notions of colimit for 2-categories, including pseudo 2-limits, lax 2-limits, and colax/oplax 2-limits. Again, by the theory of enriched categories, it follows that we can form the free cocompletion of $\mathscr K$ under pseudo 2-colimits, lax 2-colimits, or colax 2-colimits by restricting to the sub-2-category of $[\mathscr K^\circ, \mathrm{Cat}]$ closed under the respective classes of colimits of representables. However, this does not give a particularly concrete description.
Is there an explicit, simple description of the free cocompletion of a small 2-category $\mathscr K$ under (1) pseudo 2-colimits, (2) lax 2-colimits, (3) colax 2-colimits? What about when we replace "2-colimit" by "bi-colimit"?
An example of the kind of result I am hoping for is Theorem 15.18 of Garner and Shulman's Enriched categories as a free cocompletion, which gives an explicit characterisation of the free cocompletion of a locally cocomplete 2-category under lax 2-colimits of lax functors. However, I do not want to assume the existence of any local colimits.
Along similar lines, given the motivation of Cockett–Koslowski–Seely–Wood's Modules, one might conjecture that the free completion of a 2-category under lax limits of lax functors is related to their $\mathbf{trans}$.
