The behaviour of the $$overline{W}$$ -construction on the homotopy theory of bisimplicial sets

The codiagonal functor (overline{W}) transfers a Quillen closed model structure on the bisimplicial set category from the ordinary model category of simplicial sets. This bisimplicial model structure is different from the so called Moerdijk model structure, which is similarly transferred from simplicial sets but through the diagonal functor. We show the mutual relationship of these two closed model structures on the category of bisimplicial sets.