For each small 2-category $\mathscr K$, the 2-category $[\mathscr K^\circ, \mathrm{Cat}]$ of 2-functors and 2-natural transformations has a universal property: it is the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors; this follows from $\mathrm{Cat}$-enriched category theory.
We may also consider the 2-categories $[\mathscr K^\circ, \mathrm{Cat}]_p$, $[\mathscr K^\circ, \mathrm{Cat}]_l$, $[\mathscr K^\circ, \mathrm{Cat}]_c$ of 2-functors and pseudo/lax/colax natural transformations respectively. Does the Yoneda embedding of $\mathscr K$ exhibit a universal property for any of these constructions?
