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The OP mixed up 'left' and 'right' in the text below the adjoint functors: $2\text{-Cat}$ is the usual notation for strict 2-categories, so it is the *rightgoing* functor which is the strictification. I thought commenting/pinging because of this oversight too obtrusive and corrected it myself.
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edit approved Aug 16, 2017 at 11:20
Peter Heinig
edit approved Aug 16, 2017 at 11:20
Peter Heinig
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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 leftgoingrightgoing functor is the full image of the yoneda embedding, and the rightgoingleftgoing functor is the inclusion. However, I cannot find such a statement in the literature. If it is true, a reference would be appreciated.

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 leftgoing functor is the full image of the yoneda embedding, and the rightgoing functor is the inclusion. However, I cannot find such a statement in the literature. If it is true, a reference would be appreciated.

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.

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Espen Nielsen
Espen Nielsen
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On the coherence theorem for bicategories

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 leftgoing functor is the full image of the yoneda embedding, and the rightgoing functor is the inclusion. However, I cannot find such a statement in the literature. If it is true, a reference would be appreciated.