This is really a notational thing that is kind of irking me. Consider the dual group of $G=Z_n\times Z_m$, so $G$ is a finite abelian group. I can get an order $n$ homomorphism $\varphi_n$ from $G$ to $\mathbb{C}^*$. I can also get an order $m$ homomorphism $\varphi_m$ from $G$ to $\mathbb{C}^*$.
We can then consider the direct product $\langle \varphi_n\rangle\times\langle\varphi_m\rangle$. This is clearly a group. Yet I am a little confused by what the elements of this group are. Is the element $(\varphi_n,\varphi_m)\in \langle \varphi_n\rangle\times\langle\varphi_m\rangle$ still a homomorphism?
My question is based off of this question: Help with proof for problem 14 chapter 5 Dummit and Foote. and how they were able to say that $\langle \chi_1\rangle\times\cdots\times\langle\chi_r\rangle\leq \hat{G}$. It is not clear to me how we can interpret elements of the direct product as homomorphisms. Thanks!