I originally posted it on stackexchange, but did not get any comments or answer, so I try my luck here.
I am looking for the naive notion of homotopy.
For maps $f, g: A\to B$ of simplicial sets $\Delta^{op}\to Sets$, a homotopy $H$ is a simplicial map $H:A\times\Delta^{1}\to B$ such that $H|_{A\times\{0\}}=f$ and $H|_{A\times\{1\}}=g$, or equivalently a collection of maps $H_i: X_n\to Y_{n+1}, 0\le i\le n$ subject to a bunch of identities, see encyclopedia of math.
Now for maps $f, g:A\to B$ of bisimplicial sets $\Delta^{op}\times\Delta^{op}\to Sets$, what is a homotopy between them?
Is it a map $A\times\Delta^{1,1}\to B$ such that a suitable boudary condition hold? But now there are four vertexes of $\Delta^{1,1}$.
Obviously, we may define a map $A\times\Delta^{0,1}\to B$ or $A\times\Delta^{1,0}\to B$ to be vertical/horizontal homotopy.