Proposition: Restricted classical covariance

Let $F_e$ be a formal diffeomorphism equal to the identity on a neighborhood of the initial surface, with compactly supported positive-order generators. Then

Proof. The scalar wave equation is covariant under pullback, so ${\mathsf{H}}_{q_e}\circ F_e$ solves the coordinate equation for $F_e^{\ast }{q}_e$. The initial data agree because $F_e$ is the identity in an initial collar. Coefficientwise uniqueness proves the first equality. Since $F_e^{\ast }{T}_e=T_e\circ F_e$, the inverse-map identity proves the second.