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Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function f in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type f=φ g, where g is cyclic, φ is a contractive multiplier, and f=g. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors thatarXiv:2203.08179v1 fatcat:s7i7jfyiancoznzgz7fjyoakyi