Put a bit more coherently, given $p$ and $q$ as distinct prime numbers, and thus $(p,q)=1$, if
$p^{(q-1)} + q^{(p-1)} \equiv 1 \pmod p$ and $p^{(q-1)} + q^{(p-1)} \equiv 1 \pmod q,$
why does that lead to
$p^{(q-1)} + q^{(p-1)} \equiv 1 \pmod {pq}?$
The textbook I'm working with jumps to that conclusion as if it were obvious, but it's not. Not to me, at least.