As stated in the question, and I have looked at other questions on this topic here, but I am still very confused whether the equation $$ U(\Lambda,a)\hat\phi(x) U^{-1}(\Lambda,a) = R(\Lambda,a) \hat\phi(\Lambda^{-1}x + a) $$ is something that can be derived from the theory of Poincare representations, or whether it is an axiom of Quantum Field Theory?

So $\hat\phi$ denotes a quantum field operator, $U(\Lambda,a)$ an infinite dimensional unitary irrep of Poincare on the Hilbert space of particle states, and $R(\Lambda,a)$ a finite-dim irrep of Poincare acting on the corresponding classical fields $\phi(x)$, eg spinors, vectors etc.

As stated in the question, and I have looked at other questions on this topic here, I am still confused whether the equation $$ U(\Lambda,a)\hat\phi(x) U^{-1}(\Lambda,a) = R(\Lambda,a) \hat\phi(\Lambda^{-1}x + a) $$ is something that can be derived from the theory of Poincare representations, or whether it is an axiom of Quantum Field Theory?

So $\hat\phi(x)$ denotes a quantum field operator, $U(\Lambda,a)$ an infinite dimensional unitary irrep of Poincare on the Hilbert space of particle states, and $R(\Lambda,a)$ a finite-dim irrep of Poincare acting on the (components of) corresponding classical fields $\phi(x)$, eg spinors, vectors etc.