In differential geometry, pullback is a useful operation to induce various objects on a manifold to another manifold. Furthermore, pullback is very natural in a sense that it commutes with exterior derivative, Lie derivative, etc.
My question concerns the naturality of pullback in a complicated situation.
Let $M$ be a smooth $m$-manifold. Consider a complicated vector bundle $E$ over $M$. For example, let's say $$E = C^\infty(M) \oplus \Omega^1(M) \oplus \Omega^1(M) \oplus S^2(T^*M) \oplus \Omega^m(M),$$ where $S^2(T^*M)$ is the space of symmetric rank-2 tensor of $T^*M$. Let $F: E \to \Omega^m(M)$ be a map, not necessarily linear but preserves fibers. One example of $F$ could be $$F(\phi(p), \omega(p), \tau(p), g(p), \eta(p)) = (\phi(p) + g^{ab}(p) \omega_a(p) \tau_b(p)) \eta(p), \qquad p\in M.$$ After $F$ is given, we consider its integral. More precisely, define a function $S: C^\infty(M) \times \Omega^1(M) \times \Omega^1(M) \times S^2(T^*M) \times \Omega^m(M) \to \mathbb R$, given by $$S(\phi, \omega, \tau, g, \eta) = \int_M F(\phi(p), \omega(p), \tau(p), g(p), \eta(p)).$$ Here, I slightly abused notation. The RHS actually means $$\int_M (p \mapsto F(\phi(p), \omega(p), \tau(p), g(p), \eta(p))).$$
Question: Do we have $$S(f^*\phi, f^*\omega, f^*\tau, f^*g, f^*\eta) = S(\phi, \omega, \tau, g, \eta),$$ whenever $f: M\to M$ is a diffeomorphism?
I suspect that the answer should be yes, due to a sort of "naturality" of the pullback operation. However, I cannot formally prove this result.