I want to understand the following commutative diagram in class field theory (from https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.3.3.3):
Milne says this follows from the definition of Artin reciprocity map and $Inf(u_{E/K})=[L:E]u_{L/K} $ where $u_{L/K}$ is the fundamental class in $H^2(L/K)$ (i.e. element which maps to $1/n +\mathbb Z $ under the invariant map). But apparently, it does not actually follow from these (at least I don't see how it does and Milne has footnote indicating that it does not).
It seems that use of cup products is indispensable. I found one proof here: https://www.math.ucla.edu/~sharifi/algnum.pdf#theorem.10.1.11
In the beginning of the corollary the author says that homomorphism $\chi :(Gal(E/K) \rightarrow \mathbb Q/\mathbb Z $ can be viewed as homomorphism of $Gal(L/K)$ and even its abelianization (uses slightly different notation ). This much is fine. Further he says that in order to prove the commutativity of the diagram it suffices to prove $\chi (\phi_{L/K}(a)|_{E})=\chi (\phi _{E/K}(a)) $.
I do not understand why is this true?
Any help is appreciated.