On twisting functions in twisted cartesian products and twisted tensor products

For a given twisted cartesian products of simplicial sets, we construct the corresponding twisted tensor product in the sense of Brown, with an explicit twisting function whose formula is simple without using inductions. This is done by choosing an explicit morphism of topological monoids from Kan’s loop group to Moore loop spaces, following Berger’s work on simplicial prisms. We follow the choice of Brown and Berger on such a morphism, which is different from that of Gugenheim and Szczarba.