Composition theorem for randomized communication complexity
My understanding is that it's not following from [Nisan94], but from [BCW98] (note that there are two citations provided from Theorem 5), specifically their Theorem 2.1. while phrased for quantum, this generalizes to classical models as well.
See Theorem 69 of Troy Lee and Adi Shraibman's survey [LS09], available, e.g., at this address.

[BCW98] Buhrman, Harry; Cleve, Richard; Wigderson, Avi. Quantum vs. classical communication and computation. STOC '98 (Dallas, TX), 63--68, ACM, New York, 1999. Available at https://arxiv.org/abs/quant-ph/9802040
[LS09] Troy Lee and Adi Shraibman, Lower Bounds in Communication Complexity, Foundations and Trends® in Theoretical Computer Science: Vol. 3: No. 4, pp 263-399. 2009. http://dx.doi.org/10.1561/0400000040
[Nisan94] Nisan, N. The communication complexity of threshold gates. Combinatorics, Paul Erdős is eighty, Vol. 1, 301--315, Bolyai Soc. Math. Stud., János Bolyai Math. Soc., Budapest, 1993.