  The goal of this paper is to prove that the classifying spaces of categories
of algebras governed by a prop can be determined by using function spaces on
the category of props. We first consider a function space of props to define
the moduli space of algebra structures over this prop on an object of the base
category. Then we mainly prove that this moduli space is the homotopy fiber of
a forgetful map of classifying spaces, generalizing to the prop setting a
theorem of Rezk. The crux of our proof lies in the construction of certain
universal diagrams in categories of algebras over a prop. We introduce a
general method to carry out such constructions in a functorial way.
