Intersection of two deterministic parity automata

Given two deterministic parity automata $ A_1=(Q_1,\Sigma,\delta,q_{01},c_1)$ and $ A_2=(Q_2,\Sigma,\delta,q_{02},c_2)$ with the finite set of states $ Q_i$ , the finite alphabet $ \Sigma_i$ , the transition function $ \delta_i : Q_i \times \Sigma \rightarrow Q_i$ , the initial state $ q_{0i}$ in $ Q_i$ , and the coloring function $ c_i : Q_i \rightarrow \mathbb{N}$ , with $ i \in\{0,1\}$ .

How does formally is defined the intersection of two deterministic parity automata? In particular how the coloring function are combined?