Let $L/K$, $L'/K$ are finite extensions of a number field, $LL'$ be the composition field, let $p$ be a prime in $K$, $q, q'$ be primes over $p$ in $L$ and $L'$ respectively, $q''$ be the prime over $p$ in $LL'$.
Will we have completions $(LL')_{q''}\simeq L_q{L'}_{q'}$?