In the Tate spectral sequence, the edge morphism coincides with the cup product. The proof is written in Neukirch-Schmidt-Wingberg's book: Cohomology of Number Fields (Theorem 2.5.5,p125). https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/NSW2.2.pdf
However I don't understand the proof.
In (p.126, line 5), "we obtain a commutative diagram", but no explanation about the commutativity is written. I think this is not obvious, and the commutativitiy should rather be the heart of the proof. (In other words, line 9, why can the lower arrow be the evaluation map?)
It would be appreciated if you help me, explaining it or suggesting another book or article.