It is known that a morphism between principal $G$-bundles over the same base must be an isomorphism of principal bundles.
Can I ask: is it true that a morphism of principal $G$-bundles over different bases must be a pull-back diagram?
More precisely, suppose $\pi:P\to X$, $\pi^\prime:P^\prime\to X^\prime$ are principal $G$-bundles and $(F,f):(P,X)\to(P^\prime,X^\prime)$ is a $G$-bundle morphism, namely $F$ is $G$-equivariant and $\pi^\prime\circ F=f\circ\pi$. Then does it necessarily hold that $P\cong f^*P^\prime$?
$f^*P'=\{(x,y):x\in X, y\in P':\pi'(y)=f(x)\}$ define $H:P\rightarrow f^*P'$ by $H(z)=(\pi(z),F(z))$, $H$ is well defined since $\pi'(F(z))=f\pi(z))$, we deduce that $H$ is an isomorphism since it is a morphism of principal bundles.