This post is related to this post: https://math.stackexchange.com/questions/4676947/group-dual-of-product-is-isomorphic-to-product-of-group-duals#:~:text=We%20define%20the%20dual%20of,be%20two%20finite%20abelian%20groups. but I do not really understand the answer.
More specifically, this question comes from Lang Algebra. 
Here is the theorem. I just do not really get what Lang is doing. For example, I dont get how he establishes the homomorphism from B^ $\times$ C^ $\to$ (B $\times$ C)^. I understand that $(\psi_1,\psi_2) \in $ B^ $\times$ C^ , but how does $\psi_1(x) + \psi(y)$ help us? I dont see at all how this defines a homomorphism between the two. A homomorphism would have to send an element from B^ $\times$ C^ to an element of ($B \times C$)^. So we would have to send $(\psi_1,\psi_2)$ $\to$ $\psi$ where $\psi \in $ ($B \times C$)^. Could someone please help me write out the explicit isomorphism?
An element of $(B \times C)$^ is a homomorphism from $B \times C$ to $Z_m$ where $m$ is the exponent of A (this is how lang defines it). Ive been stumped on this theorem for 2 hours (embarrassingly) and i feel like im missing something obvious since i cant find anything about it online really. The stack exchange post i linked just basically says exactly what Lang writes, so that doesnt help in my case. Any help is appreciated.
