Induced map on associated bundles

Let $P\to M$ and $P'\to M'$ be principal G-bundles . A principal bundle morphism consists of a smooth map smooth map $f:M\to M'$ and a smooth map $\Phi:P\to P'$ such that

Let $F$ and $F'$ be left $G$-spaces, and let $\psi:F\to F'$ be $G$-equivariant.

Construction. Define

This is well-defined (independent of the representative $(p,u)$) and is a smooth bundle map covering $f$:

When $F=F'$ and $\psi=\mathrm{id}$, one gets the induced map on a fixed associated bundle functorially from $\Phi$.

Examples

1. (Pullback trivialization maps) If $P'=P$ and $f=\mathrm{id}_M$, then a gauge transformation $\Phi$ induces a bundle automorphism of any associated bundle $P\times_G F$ by $[p,u]\mapsto [\Phi(p),u]$.
2. (Vector bundle case) For $F=V$ and $F'=V'$ linear $G$-spaces, a $G$-equivariant linear map $\psi:V\to V'$ yields a vector bundle morphism $P\times_G V\to P'\times_G V'$ covering $f$.
3. (Adjoint functoriality) Taking $F=F'=G$ with conjugation and $\psi=\mathrm{id}$ gives a morphism of adjoint bundles induced from any principal bundle morphism.