The photon magnetic moment has not a perpendicular component and is fully paramagnetic

Our paper Phys. Rev. D $\mathbf{\text{79}}$, 093002 (2009), in which it was shown the paramagnetic behavior of photons propagating in magnetized vacuum, is criticized in Phys. Rev. D $\mathbf{\text{81}}$, 105019, (2010) and even claimed that the photon has a diamagnetic component. Here it is shown that such criticism is inadequate and that the alleged “perpendicular component” is due to a mistake in differentiating a vanishing term with regard to the magnetic field $B$, or either by mistaking the derivative of a scalar product as that of a dyadic product. A discussion on the physical side of the problem is also made.