Faraday's Law From two different Frame of Reference

Let's take Faraday's experiments of induce emf where He take a loop and a magnetic field, for instance let's suppose we have an coil and a magnet, Now If we move the magnet towards the coil there will be an induce electric field and Thus the force $$\vec{F}=q\vec{E}$$ on charges of loop and thus the induce electric field but Now suppose I'm in frame of reference that is moving along with magnet with the same velocity as the magnet so that magnet is rest in my frame but coil is moving, in this case I will say charge in coil is moving in magnetic field and thus there is an Lorentz force $$\vec{F}=q(\vec{v}\times\vec{B})$$ and thus the induce emf. So How the same phenomenon is explained from two different Frame of reference with two different principals? Or my reasoning is wrong? Is it possible that a same phenomenon( as here) In general, Can be explained with two different principals from two different frame of reference?