TITLE: On the coherence theorem for bicategories
QUESTION [12 upvotes]: The coherence theorem for bicategories, as usually stated, reads

Any bicategory $B$ is biequivalent to a (strict) 2-category.

It is possible to give an explicit construction of the strictification as the full image of its Yoneda embedding $y:B\rightarrow [B,\text{Cat}]$, see for instance this reference.
This seems like a natural construction, so I would expect an equivalence of tricategories
$$ \text{Bicat} \leftrightarrows 2\text{-Cat}$$
where the rightgoing functor is the full image of the yoneda embedding, and the leftgoing functor is the inclusion. However, I cannot find such a statement in the literature. If it is true, a reference would be appreciated.

REPLY [12 votes]: Probably you had some trouble finding this because the search term $2$-$\text{Cat}$ is not accurate enough; you want not the cartesian monoidal product on $2$-$\text{Cat}$, but rather what is called the Gray monoidal product; the tricategory you want then is denoted $\text{Gray}$, the tricategory of strict 2-categories, strict 2-functors, pseudonatural transformations, and modifications between them, but most usefully considered as equipped with the Gray tensor product. 
Anyway, see this paper by Steve Lack: Bicat is not triequivalent to Gray. This actually corrects a slight misstatement in the monograph by Gordon-Power-Street. See the end of Lack's paper for the punchline. I'm not sure what the state of the art is, but perhaps you could email Lack (or perhaps Street).