On the equivalence of two approaches to problems of quasiconformal analysis

We prove that the two approaches to describing homeomorphisms in modern quasiconformal analysis are equivalent: A homeomorphism changes under control the capacity of the image of a condenser in terms of the weighted capacity of a condenser in the preimage if and only if the modulus of the image of a family of curves can be estimated in terms of the weighted modulus of the original family of curves.