  We show that the functor that takes a multicosimplicial object in a model
category to its diagonal cosimplicial object is a right Quillen functor. This
implies that the diagonal of a Reedy fibrant multicosimplicial object is a
Reedy fibrant cosimplicial object, which has applications to the calculus of
functors. We also show that, although the diagonal functor is a Quillen
functor, it is not a Quillen equivalence for multicosimplicial spaces.
  We also discuss total objects and homotopy limits of multicosimplicial
objects. We show that the total object of a multicosimplicial object is
isomorphic to the total object of the diagonal, and that the diagonal embedding
of the cosimplicial indexing category into the multicosimplicial indexing
category is homotopy left cofinal, which implies that the homotopy limits are
weakly equivalent if the multicosimplicial object is at least objectwise
fibrant.
