Character of tensor product of linear representations is product of characters

Statement for a field
Suppose $$G$$ is a group (not necessarily finite) and $$\alpha, \beta$$ are finite-dimensional linear representations of $$G$$ over a field $$K$$. Denote by $$\chi_\alpha, \chi_\beta$$ respectively the characters of $$\alpha$$ and $$\beta$$. Denote by $$\alpha \otimes \beta$$ the tensor product of linear representations $$\alpha$$ and $$\beta$$ and by $$\chi_{\alpha \otimes \beta}$$ its character. Then, for any $$g \in G$$, we have:

$$\chi_{\alpha \otimes \beta}(g) = \chi_\alpha(g)\chi_\beta(g)$$

In point-free notation, this states that $$\chi_{\alpha \otimes \beta} = \chi_\alpha\chi_\beta$$.