Let $C_m$ denote the cylic group of order $m$. Recall $\widehat{C}_m=\{ \psi_j: j\in C_m\},$ where $\psi_j$ is defined by \begin{align} \psi_j(k)=e\Big( \frac{2\pi \sqrt{-1}jk}{m}\Big). \end{align} For any product $C_{m_1}\times \cdots C_{m_{\ell}}$, there is a canonical isomorphism \begin{align*} \widehat{ C_{m_1}\times \cdots \times C_{m_{\ell}} } \to\widehat{C}_{m_1}\times \cdots \times \widehat{C}_{m_{\ell}} \end{align*} where $\psi_{(j_1, \ldots, j_{\ell})}$ can be identified with \begin{align*} \psi_{j_1}\psi_{j_2}\cdots \psi_{j_{\ell}}. \end{align*}
Suppose $G$ is a finite abelian group. I would like to describe $\widehat{G}$ as follows. Choose an isomorphism of abelian groups \begin{align*} \phi: G\to C_{m_1} \times C_{m_2} \times\cdots \times C_{m_t}, \end{align*} where the $m_i$ are distinct and $m_1\mid m_2\mid \cdots \mid m_t$. There is an induced isomorphism \begin{align*} \Phi:\widehat{G}\to \widehat{C}_{m_1}\times \cdots \times\widehat{C}_{m_t} \end{align*} in which $\psi_g\in \widehat{G}$ is identified with \begin{align} \psi_{\phi(g)}\equiv \psi_{\pi_1\big(\phi(g)\big)}\,\psi_{\pi_2\big(\phi(g)\big)}\cdots \psi_{\pi_t\big(\phi(g)\big)}, \end{align} where $\pi_i: C_{m_1}\times\cdots \times C_{m_t}\to C_{m_i}$ denotes projection. Finally, we declare \begin{align} \psi_g(h)\equiv \psi_{\phi(g)}\big(\phi(h)\big) \end{align} for each $g,h\in G$.
Questions
Is there a more reasonable way to think of $\widehat{G}$ ?
This description depends on a choice of isomorphism $\phi$. But if another isomorphism is chosen, then can the two resulting descriptions be related in anyway?
Thanks!