The geometric realization functor, \rar{|?|_T}{\ss}{KTop}, 
is known to commute with finite limits if the
reflection of \m{T[1]} into the category of \m{T_0}-spaces is a
Hausdorff space, where \m{T} is a cosimplicial \m{k}-space. We
have previously shown that for the categories \m{Fco}, \m{ConsFco},
\m{Con}, \m{Lim}, \m{PsT}, \m{Born}, and \m{PreOrd}, the
initiality of the inclusion of the boundary \m{\dot Y_{0n}} of
\m{Y_{0n}} into \m{Y_{0n}} guarantees the commutation of finite
limits by the geometric realization functor, \rar{\hs{1}|?|_Y}{\ss}{A}. Here we
show that in the above mentioned categories the initiality
condition may be viewed as a generalized Hausdorff condition on
the interval \m{Y_{01}}, as is the case in the classical situation
where \m{A} is the category \m{KTop}.