I have a question concerning the proof, that the scalar multiplication of a module is well-defined. I guess, that the universal property of the tensor product will help, but I am not quite sure in which way.
Consider the left $L$-vector spaces $A:=\bigoplus_{\sigma \in G} L u_\sigma, B:= \bigoplus_{\sigma \in G} L v_\sigma, C:= \bigoplus_{\sigma \in G}L w_\sigma$ and the tensor product $M:=A \dot{\otimes}_L B$ (with the multiplication $la \dot{\otimes}b = a \dot{\otimes}lb$ so there is a module structure). Now I have to proof, that the scalar multiplication $C \times M \to M$ $$ l w_\sigma (a \dot{\otimes}b) := l u_\sigma a\dot{\otimes} v_\sigma b$$ is well-defined making $M$ a $C$-module
I know, that the notation is kind of complicated, so a general idea how to do it generally and how to include the universal property would be greatly appreciated
Thanks in advance