I want to know how does someone usually go, that is, what is the canonical way to go proving that having two notions of forcing $P$ and $Q$, the respective generic extensions $P[G]$ and $Q[H]$ are equivalent.
More specific, how would the proof of: forcing a Cohen real $\kappa$ many times and forcing $\kappa$ Cohen reals give the same forcing extension go?
Is it by showing that the complete Boolean algebras associated are the same by proving either $P$ is dense on $B(Q)$ or $Q$ is dense in $B(P)$? But doesn't that require too much information about these completions?
Or is it by finding an automorphism between the generics? Any automorphism will do?
Thank you for your time, it is my first question and I am not sure if it is appropriate I hope it's not too basic, but I am seriously having problems understanding this.