You cannot really formulate such a theorem at all.
Fixed points are for self-maps (i.e. $f:X \to X$ for some $X$).
The FPP (fixed point property) i.e. "every continuous self-map of $X$ has a fixed point" is a topological property. So any space $Y$ homeomeomorphic to $X$ has it when $X$ has it and vice versa.
This is proved easily by "map transportation": if $f: Y \to Y$ is a continuous self map and $h: X \to Y$ is a homeomorphism, then $h^{-1} \circ f \circ h$ is a continuous self map of $X$, so has a fixed point $p \in X$ and then $h(p)$ is one for $f$ (all this is simple to check). So your $D'^2$ does have the FPP. Maybe that is what you really want?