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 is the intersection of two deterministic parity automata formally defined? In particular, how are the coloring functions combined?

• Hint: you can start by proving that there is no way to define a correct coloring function for the intersection on the states $Q_1\times Q_2$ of the product automaton. – Denis Jun 20 '19 at 15:21