According to the article The homotopy infinite symmetric product represents stable homotopy theory, Proposition 4.5 states:
There is an equivalence of topological monoids $$\pi:SP_{h}(X)\xrightarrow{\thicksim} \mathrm{\Gamma}^{+}(X)$$ for any pointed CW-complex $X$.
The autor, Schlichtkrull, asserts we may work over simplicial sets throughout that paper (actually, he works over topological spaces), so I would like to prove this proposition in its simplicial version. At some point of the proof, $X$ is supposed to be the geometric realization of a pointed simplicial set $X_{\bullet}$ as a particular case
then the general case follows from this.
Nevertheless, what is the trick when $X\in\mathbf{sSet}_{\ast}$? (I mean what is the analogous step at this point).
Thanks.