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 $H^2(\Gamma_{L/K}, L^\times)$.