# Universal property of category of categories

As discussed here, Using the universal property of spaces, the $$(\infty,1)$$-category of spaces has a universal property: it is the free $$\infty$$-categorical cocompletion of the terminal category $$*$$. That is, for any $$(\infty,1)$$-category $$C$$ with all colimits, there is an equivalence $$Fun^{cocontinuous}(Spaces, C) \cong Fun(*,C) \cong C$$ I am wondering if there is a similar universal property for $$Cat^{(\infty,1)}$$, the $$(\infty,2)$$-category of $$(\infty,1)$$-categories.

Note that $$Cat^{(\infty,1)}$$ has all lax colimits and any category $$C$$ is the lax colimit of the functor $$C \rightarrow * \rightarrow Cat^{(\infty,1)}$$. So one guess is that $$Cat^{(\infty,1)}$$ is the lax cocompletion of the terminal category $$*$$ in the sense that for any $$(\infty,2)$$-category $$C$$ with lax colimits, there is an equivalence $$Fun^{Lax \ cocontinuous}(Cat^{(\infty,1)}, C) \cong Fun(*, C) \cong C$$ Does this proposed equivalence hold and, if so, has it been studied somewhere?

• "the category of spaces" here being the $(\infty,1)$-category of (homotopy types of) spaces, and cocompletion is homotopy cocompletion. Obviously you know this, but leading with the abuse of terminology is not helpful. Commented Jul 10 at 3:17
• There should be a result along the lines that if $V$ is a cocomplete closed symmetric monoidal ($\infty$?)-category then $V$ is the free cocomplete $V$-enriched category on a point, a special case of the universal property of presheaf categories. Morally we'd like to apply that result to $V = \text{Cat}$ here. Commented Jul 10 at 3:57