  The equation $f^n+g^n=1$, $n\in\mathbb{N}$ can be regarded as the Fermat
Diophantine equation over the function field. In this paper we study the
characterization of entire solutions of some system of Fermat type functional
equations by taking $e^{g_1(z)}$ and $e^{g_2(z)}$ in the right hand side of
each equation, where $g_1(z)$ and $g_2(z)$ are polynomials in $\mathbb{C}^n$.
Our results extend and generalize some recent results. Moreover, some examples
have been exhibited to show that our results are precise to some extent.
