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?