Let $L$ be a Lie algebra with a basis $(x_i:i\in \Omega)$, and $\mathfrak{U}(L)$ the universal enveloping algebra of $L$.
Let $\mathfrak{S}$ denote the symmetric algebra in the variables $z_i$ ($i\in \Omega$).
The essential point in PBW theorem is that given a total order on set $\Omega$, for each non-decreasing sequence $i_1\le i_2\le \cdots \le i_k$ of elements of $I$, the collection of elements $x_{i_1}x_{i_2}\cdots x_{i_k}$ together with $1$ forms a basis of $\mathfrak{U}(L)$. Also, $\mathfrak{U}(L)$ and $\mathfrak{S}$ are isomorphic as associative algebras.
However, the proof of PBW theorem in Humphreys' Lie algebra is based on following Lemma. My question, what is the intuition to consider such family of maps $f_m$, and why the domain is considered to be $L\otimes S_m$? [Because the proof of PBW theorem in Jacobson's book is a simple commutator calculus based computation.]
