The well known nerve functor from small categories to simplicial sets has a left adjoint, namely the fundamental category functor. Does the double nerve functor $N^2:2Cat\rightarrow sSSet$ from 2-categories to bisimplicial sets have a similar left adjoint? I think I've seen a vague reference to it somewhere but nothing explicit. 



Left adjoint to Double Nerve?
Yes, by the adjoint functor theorem, because it preserves all limits and both categories are locally presentable. It's also an instance of the general notion of [nerve and realization](https://ncatlab.org/nlab/show/nerve+and+realization) determined by a canonical cobisimplicial 2-category.


