Lemma

The mapping $\lambda$ is a homeomorphism.

The mapping $\lambda_0$ is surjective and two elements of $(B'\times_BE)\times F$ have the same image under $\lambda_0$ if and only if they belong to the same orbit for the operation of G in $(B'\times_BE)\times F$. It follows that the mapping $\lambda$ is bijective. Since the mapping $\pi$ is open, the same is true of the mapping Id$_{B'}\times_B\pi ($I, p. 17, prop. 8), hence of $\lambda_0$ and of $\lambda$ (TG, I, p. 32, prop. 3), which proves that $\lambda$ is a homeomorphism.