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.](https://math.berkeley.edu/~honigska/coherence_essay.pdf)


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.



On the coherence theorem for bicategories
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](http://arxiv.org/pdf/math/0612299.pdf). 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). 


