  We introduce the concept of morphism of pseudogroups generalizing the
\'etal\'e morphisms of Haefliger. With our definition, any continuous foliated
map induces a morphism between the corresponding holonomy pseudogroups. The
main theorem states that any morphism between complete Riemannian pseudogroups
is complete, has a closure and its maps are \cinf\ along the orbit closures.
Here, completeness and closure are versions for morphisms of concepts
introduced by Haefliger for pseudogroups. This result is applied to approximate
foliated maps by smooth ones in the case of transversely complete Riemannian
foliations, yielding the foliated homotopy invariance of their spectral
sequence. This generalizes the topological invariance of their basic
cohomology, shown by El Kacimi-Nicolau. A different proof of the spectral
sequence invariance was also given by the second author.
