I'm reading the book "Cohomology of Number fields" by Neukirch(Proposition 1.4.3) in which I am not unable to follow one statement:
Let $0 \rightarrow A^{'} \rightarrow A \rightarrow A^{''} \rightarrow 0$ and $0 \rightarrow C^{'} \rightarrow C \rightarrow C^{''} \rightarrow 0$ be exact sequences of $G$ modules. A pairing of $G$ modules is bilinear map $$\theta:U \times V \rightarrow W$$ such that $\theta(\sigma u ,\sigma v)=\sigma \theta( u , v)$ for all $\sigma \in G$ and where $U$ and $V$ are $G$ modules. Let $B$ be another $G$ module.
What do we mean by the statement " Suppose we are given a pairing $\psi: A\times B \rightarrow C $ which induces pairing $ A^{'}\times B \rightarrow C^{'}$ and $ A^{''}\times B \rightarrow C^{''}. $ " Whether here he is assuming $\psi(A^{'}\times B)= 0?$