Principal bundle morphism

Related: vector bundle morphism.
Let $P (G, π, B)$ and $P^{'} (G^{'}, π^{'}, B^{'})$ be two principal $G$ -bundles with base spaces $B$ and $B^{'}$ , respectively. Let $h : G \to G^{'}$ be a group homomorphism between the structure groups $G$ and $G^{'}$ . A morphism $f : P \to P^{'}$ between these principal bundles is a pair of smooth maps $(f_{B}, f_{P})$ , where $f_{B} : B \to B^{'}$ and $f_{P} : P \to P^{'}$ , satisfying the following conditions:

The diagram

\begin{array}{ccc} P & \overset{f_{P}}{⟶} & P^{'} \\ π ↓ & ↓ π^{'} \\ B & \overset{f_{B}}{⟶} & B^{'} \end{array}

commutes, i.e., $π^{'} \circ f_{P} = f_{B} \circ π$ .
2. The map $f_{P}$ is $G$ -equivariant with respect to $h$ , that is:

f_{P} (p \cdot g) = f_{P} (p) \cdot h (g), \forall p \in P, g \in G .

Here, $f_{P} (p \cdot g)$ and $f_{P} (p) \cdot h (g)$ are elements of $P^{'}$ , and $h (g) \in G^{'}$ .