It is shown that two continuous-time control systems are dynamically feedback equivalent if and only if their Euler's discretizations are h-dynamically feedback equivalent for every discretization step h. In particular, a continuous-time system is dynamically feedback linearizable if and only if its Euler's discretization is h-dynamically feedback linearizable for every h > 0. The proofs of these results are based on algebraic characterizations of dynamic feedback equivalence for continuous-time and discrete-time systems. Two continuous-time systems are dynamically feedback equivalent if and only if their differential algebras are isomorphic. Similarly, two discrete-time systems are dynamically feedback equivalent if and only if their difference algebras are isomorphic. Differential algebras corresponding to continuous-time systems and difference algebras corresponding to discretizations of those systems form two categories. Discretization induces a covariant functor from one category to the other. This functor may be inverted as the difference algebras are equipped with the whole family of difference operators corresponding to all discretizations steps h.