In Top, *how* do conjugate homorphisms of groups induce homotopies of classifying spaces? 
They exist— there's an abstract proof, but how is BG → BH written in terms of the homotopy?? 

If G and H are only H-spaces, a homomorphism induces a  map of a projective plane. In Top, *how* do conjugate H-maps  induce homotopies of  maps of  projective planes??