Fixed point algebra of the tensor product of ultragraph $C^{*}$-algebras

We prove that the $C^*$-algebra of the Cartesian product of ultragraphs $\mathcal G_1$ and $\mathcal G_2$ is isomorphic to a certain subalgebra $\mathcal A$ of the tensor product $C^*(\mathcal G_1)\otimes C^*(\mathcal G_2)$. When $\mathcal G_1$ and $\mathcal G_2$ are row-finite without sinks, the subalgebra $\mathcal A$ is isomorphic to the fixed point algebra with respect to the gauge action of the unit circle $\mathbb T$ on $C^*(\mathcal G_1)\otimes C^*(\mathcal G_2)$.