This follows from the fact that retractions are type-preserving, because they are defined using isomorphisms between apartments. (See the argument used in the proof of (3.16) in Suzuki's book.)
In particular, in the case $m=0$ where $E \subset C$, you already know that $\phi(\phi'(C)) = C$, so $\phi(\phi'(E)) \subset C$. The only subset of $C$ of the same type as $E$ is $E$ itself, so you are done.