ct.category theory - Is there a natural way to give a bisimplicial structure on a 2-category?

I mean by the nerve functor.

Given a 2-category $mathcal{C}$, if we forget the 2-category structure (just view $mathcal{C}$ as a category), the nerve functor will give us a simplicial set $Nmathcal{C}$. However, $mathcal{C}$ is a 2-category, thus for any two objects $x,yinmathcal{C}$, $Hom_{mathcal{C}}(x,y)$ is a category, applying the nerve functor gives us a simplicial set $N(Hom(x,y))$.

My question is, can these two simplicial set structure compatible in some way, gives us a bisimplicial set $N_{p,q}(mathcal{C})$, say? Or is there another way to give a bisimplicial structure on a 2-category?