Invariants of spaces over some metric space

We consider the category of locally connected topological spaces over some complete metric space \(M\), whose objects are continuous functions to \(M\) and whose morphisms between two given functions \(f \colon X \rightarrow M\) and \(g \colon Y \rightarrow M\) are the continuous maps \(\varphi \colon X \rightarrow Y\) such that \[ \xymatrix{ X \ar[rr]^{\varphi} \ar[dr]_f & & Y \ar[dl]^g \\ & M } \] commutes. In the following we will consider several invariants (mostly given as functors to other categories and mostly in the special case where \(M = {\mathbb{R}}\)) under isomorphisms of objects in this category.