Recall that the extension of Fermat's Little Theorem says that $a^{\phi(n)} \equiv 1 \pmod{n}$ if $\gcd(a,n) = 1$.

What can you say about $m^{\phi(n)} + n^{\phi(m)} \pmod{n}$ and $m^{\phi(n)} + n^{\phi(m)} \pmod{m}$?