Let $G$ be a Lie group, let $\pi: P \to M$ be a smooth principal $G$-bundle, and let $f: N \to M$ be a smooth map of smooth manifolds. Form the pullback principal $G$-bundle $f^*P = \{(x,p) \in N \times P : f(x) = \pi(p)\}$ over $N$, with bundle projection $\operatorname{pr}_N: f^*P \to N$, projection $\operatorname{pr}_P: f^*P \to P$, and right $G$-action $(x,p)\cdot g = (x,p\cdot g)$.

Let $\rho: Q \to N$ be a smooth principal $G$-bundle and let $H: Q \to P$ be a smooth $G$-equivariant map such that $\pi \circ H = f \circ \rho$. Then there exists a unique smooth $G$-equivariant map $\widetilde H: Q \to f^*P$ satisfying $\operatorname{pr}_N \circ \widetilde H = \rho$ and $\operatorname{pr}_P \circ \widetilde H = H$. Equivalently, $\widetilde H$ is the unique principal bundle morphism from $Q \to N$ to $f^*P \to N$ over $\operatorname{id}_N$ whose composition with $\operatorname{pr}_P$ is $H$.