Weak lensing post-Born effects are equivalent to pure lens-lens couplings

We show that the so-called post-Born effects of weak lensing at 4th order are equivalent to lens-lens couplings in the Born Approximation. We demonstrate this by explicitly showing the equivalence of the canonical weak lensing approach at 4th order using the anisotropy remapping method, to that of the 4th order calculation of the lens-lens coupling effects using the Boltzmann equation approach that was first developed in [Phys. Rev. D89, 123006]. Furthermore, we argue that to incorporate true post-Born effects, i.e. taking into account non-straight photon paths, require the addition of a photon deflection term which has not been taken into account in the canonical formalism nor the Boltzmann method.