Theorem 34.6:

The inclusion maps $S^{\;\cal U}_n(X)\longrightarrow S_n(X)$ ( $n = 0, 1, 2,\dots$) define a chain map of complexes. Further, these inclusion maps induce isomorphisms in homology:

$$\begin{CD} H_n^{\;{\cal U}}(X) @> \cong >> H_n(X), \quad n = 0, 1, 2, \dots \end{CD}$$