Answer of James S. Milne: Most probably, this homomorphism $$\lambda\colon\, H^{-1}(\Gamma_{L/K}, Y)\overset\sim\longrightarrow H^1(\Gamma_{L/K}, Y\otimes_{\Bbb Z} L^\times)$$ is just the cup-product with the fundamental class in $H^2(\Gamma_{L/K}, L^\times)$.