The classic geometric realization functor |-|: S^��op��KTop, where
S is the category of sets, S^��op
is the category of simplicial sets, and KTop is the category
of compactly generated Hausdorff topological spaces, is generalized to the functor
|-|Y:S^��op �� A, where A is a category geometric over S via f and Y forms a
discrete fibration over f^*��op, in Cat(A), via g. It is shown that, under certain assumptions
on A, f and g, this generalized functor commutes with finite limits if the collection
of the inclusions of the boundary Y^'_0n of Y_0n into Y_0n is strongly initial. It is further
shown, for certain geometric categories A over sets, in particular for the categories Fco,
ConsFco, Con, Lim, PsT, Born, and PreOrd, that initiality of the inclusion of the
boundary Y^'_0n of Y_0n into Y_0n guarantees commutation of the geometric realization
functor and finite limits.