I suppose the following definition is standard. Given a principal $G$-bundle $P(M,G)$ ($G$ acting freely on $P$ on the right) over some manifold $M$, an automorphism $j$ of $P$ is simply a bundle map satisfying $$ \forall u\in P, a\in G\qquad j(ua)=j(u)j'(a) $$ with $j':G\to G$ an isomorphism.
But when I read Kobayashi & Nomizu, Foundations of Differential Geometry, Vol.1 page 105, it seems the underlined equality should depend on the fact that $j'$ is taken to be the identity map. Am I missing something from the context?
