Problem
Problem 5.16 A morphism of principal bundles $\pi: P \rightarrow M, \pi^{\prime}: P^{\prime} \rightarrow M^{\prime}$, with structure groups $G, G^{\prime}$, respectively, is a smooth map $\Phi: P \rightarrow P^{\prime}$ and a Lie group homomorphism $\gamma: G \rightarrow G^{\prime}$ such that
$$
\Phi(u \cdot g)=\Phi(u) \cdot \gamma(g), \quad u \in P, g \in G
$$
(i) If $\Phi: P \rightarrow P^{\prime}$ is a morphism, then prove that there exists a unique smooth map $\phi: M \rightarrow M^{\prime}$ making the following diagram commutative:
(ii) Suppose now that $M^{\prime}=M, G^{\prime}=G, \pi=\pi^{\prime}$, and $\gamma=\mathrm{id}_{G}$. Then a principalbundle morphism $\Phi: P \rightarrow P$, is said to be an automorphism if $\Phi$ is a diffeomorphism.
Prove that all the automorphisms constitute a group with respect to composition. This group is denoted by Aut $P$. Prove that the map Aut $P \rightarrow$ Diff $M, \Phi \mapsto \phi$, with $\phi$ given as in (i) above, is a group homomorphism.
Remark The kernel of the last homomorphism is called the gauge group of $P$ and it is denoted by $\operatorname{Gau} P$.
(ii) Suppose now that $M^{\prime}=M, G^{\prime}=G, \pi=\pi^{\prime}$, and $\gamma=\mathrm{id}_{G}$. Then a principalbundle morphism $\Phi: P \rightarrow P$, is said to be an automorphism if $\Phi$ is a diffeomorphism.
Prove that all the automorphisms constitute a group with respect to composition. This group is denoted by Aut $P$. Prove that the map Aut $P \rightarrow$ Diff $M, \Phi \mapsto \phi$, with $\phi$ given as in (i) above, is a group homomorphism.
Remark The kernel of the last homomorphism is called the gauge group of $P$ and it is denoted by $\operatorname{Gau} P$.