Let $P(G, \pi, B)$ and $P'(G', \pi', B')$ be two principal $G$-bundles with base spaces $B$ and $B'$, respectively. Let $h: G \rightarrow G'$ be a group homomorphism between the structure groups $G$ and $G'$. A morphism $f: P \rightarrow P'$ between these principal bundles is a pair of smooth maps $(f_{B}, f_{P})$, where $f_{B}: B \rightarrow B'$ and $f_{P}: P \rightarrow P'$, satisfying the following conditions:
1. The diagram
$$ \begin{array}{ccc} P & \overset{f_{P}}{\longrightarrow} & P' \\ \pi \downarrow \quad & & \quad \downarrow \pi' \\ B & \overset{f_{B}}{\longrightarrow} & B' \end{array} $$commutes, i.e., $\pi' \circ f_{P} = f_{B} \circ \pi$.
2. The map $f_{P}$ is $G$-equivariant with respect to $h$, that is:
$$ f_{P}(p \cdot g) = f_{P}(p) \cdot h(g), \quad \forall p \in P, \; g \in G. $$Here, $f_{P}(p \cdot g)$ and $f_{P}(p) \cdot h(g)$ are elements of $P'$, and $h(g) \in G'$.
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Author of the notes: Antonio J. Pan-Collantes
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