Composition theorem for randomized communication complexity I am currently organizing the literature of composition theorem, and I found the paper by https://www.research.cs.rutgers.edu/~troyjlee/Composition.pdf, in their theorem 5, I find
$$ R_{1/4} (f \circ g^{n}) = \mathcal{O}(RT_{1/4}(f)R_{1/4}(g)\log RT_{1/4}(f)) $$ coming from Nisan's paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.57.6891&amp;rep=rep1&amp;type=pdf, where $RT(f)$ is denoted as query complexity and $R(f)$ is the randomized communication complexity.
However, I am not sure I notice such a conclusion has ever been drawn in Nisan's paper. The closest claim I suppose is the lemma 3, where Nisan claims that

Let $G$ be a family of functions. If a function $f$ can be computed by a circuit consisting of $s$ gates from $G$ then, for all $k$, $R^{k}(f) \leq s \cdot R^{k}_{1/3s}(G)$.

My question is : is it the lemma leading to the theorem 5 of the first paper, and how? Or am I missing something here?
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.