Proof:

Suppose that $X$ has the fixed point property and $h : X\longrightarrow Y$ is a homeomorphism. Let $g : Y\longrightarrow Y$ be an arbitrary continuous map. Applying the fixed point property to the map $f = h^{-1}\circ g\circ h$ we get a point $p \in X$ such that $f(p) = p$. The fixed point of $g$ is seen to be $h(p)$.