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The classical straightening theorem as proved by Douady and Hubbard showsthat a polynomial-like sequence is hybrid equivalent to a polynomial. Wegeneralize this result to non-autonomous iteration where one considerscomposition sequences arising from a varying sequence of functions. In order todo this, new techniques are required to control the distortion andquasiconformal dilatation of the hybrid equivalence. In particular, theCaratheodory topology for pointed domains allows us to specify the appropriatebounds on the sequence of sets on which the polynomial-like mapping sequence isdefined and give us good estimates on the degree of distortion andquasiconformality.

A Straightening Theorem for non-Autonomous Iteration