I am studying Wightman axioms and Haag–Kastler axioms for QFT from Haag's book "Local Quantum Physics". In both axiomatic frameworks, he introduces the "Time-slice Axiom" (axiom G) as
"There should be a dynamical law which allows one to compute fields at an arbitrary time in terms of fields in small time slice $\mathcal{O}=\{ x:|x^0-t|<\epsilon \}$."
I don't understand what is the difference between this axiom and the axiom concerning the transformation properties (axiom D) of the Wightman fields or local algebras. Doesn't the unitary operator $U(a,\mathbb{1})$ introduced in axiom D work as the dynamical law mentioned in axiom G?
If you know some old post or some book where I can find this answer and I didn't see please notify me.