= Solution
Define $\widetilde f:X/\mathcal F\to Y$ by $\widetilde f([x])=f(x)$. Its definition is <independent> of the representative because the classes are precisely the fibres of $f$. It is onto since $f$ is onto, and <injective> because two points with equal $f$-values lie in the same fibre.
The relation $\widetilde f\circ p=f$ and part (iii) imply <continuity> of $\widetilde f$. Since $X$ is <compact> and $p$ is <continuous>, $X/\mathcal F$ is <compact>. A <continuous> <bijection> from a <compact> space to a <Hausdorff space> is a <homeomorphism>. Therefore
$$
\boxed{X/\mathcal F\cong Y.}
$$
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