Let $G$ be an abelian group and let $X$ be a finite set. For a given group $T$, let $T^X$ be the group of functions $X\to T$ where the operation is defined pointwise. Is it true that $G\otimes\mathbb Z^X\cong G^X$?
I know that all elements of $G\otimes\mathbb Z^X$ may be written uniquely as sums $\sum_{x\in X}g_x\otimes e_x$ where $g_x \in G$ for all $x\in X$ and $e_x$ is the canonical basis element of $\mathbb Z^X$ which is $1$ at $x$ and $0$ elsewhere, and those sums look like "formal linear combinations with coefficients in $G$" which is essentially what $G^X$ is. So it seems that the map $G^X\to G\otimes\mathbb Z^X$ given by $f\mapsto \sum_{x\in X} f(x)\otimes e_x$ should indeed be an isomorphism. Is this correct?