A bicategory ($=$ weak 2-category) is a 0-cell in what category?

Question

Question.

What, in your opinion, should replace the "?" in:

If (category)$=$(0-cell in CAT), then (bicategory)$=$(0-cell in ?).

Remarks.

• "bicategory" as usual means "weak 2-category".
• I am well aware that this is sort-of-opinion-based and does not have an uncontroversial precise answer, and is not asking much more than for mathematicians more experienced in category theory to suggest
• • what the category of all bicategories should mean precisely,
• • how the category of all bicategories should be denoted.

• Still, I think there is some justification for asking this. It is useful in some contexts to be able to think of "category" as "0-cell in the category of categories", and there is use, too, for being able to do the same for "bicategory" instead of "category".

• There are some issues what definition of "functor between two bicategories" to adopt, in the first place, but I will not get into this here (see the next remark). It can be argued that the definition should at least
• • make $\mathsf{CategoryOfBicategories}$ have a all finite products.
• I do not give explanations of technical terms since I am curious what others think should be the "canonical" answer here.

Answer 1

Since it seems too complicated for this question to first ask the commenter whether they would like to make the comment an answer, I'll write one here myself (thanks to Alexander Campbell for confirming that $\mathsf{Bicat}$ is a usual notation): arguably the most usual $\mathsf{CategoryOfBicategories}$ is the category which is denoted $\mathsf{Hom}$ in Street's 1980 article in Cahiers de topologie et géométrie différentielle catégoriques, but the most usual notation nowadays is $\mathsf{Bicat}$. Subscripted variants thereof exist, to distinguish some-sort-or-full-subcategories of $\mathsf{Bicat}$, like e.g. $\mathsf{Bicat}_s$ and $\mathsf{Bicat}_l$ mentioned in a comment above, or $\mathsf{Bicat}_2$ in the sense of R. Garner, N. Gurski: The low-dimensional structures formed by tricategories. Math. Proc. Camb. Phil. Soc. 146 (2009),pp. 551--589.

And, to give one application of the usual notation, and another usual notation (namely $\mathsf{2Cat}$ for strict 2-categories), I mention that there is an adjunction

$$ \array{ & & & & \xrightarrow[]{\quad\mathrm{strictification}\quad} \hspace{-10pt} & & & & \\ & & {\huge\mathsf{Bicat}} & & \perp & & & {\huge\mathsf{2Cat}} & \\ & & & &\xleftarrow[\quad\mathrm{inclusion}\quad]{} \hspace{-10pt} & & & &\qquad. \\ }$$