There is a double iteration of the last vertex map yielding $Sd^2\Delta^n \to \Delta^n$. My question is whether this is a strong homotopy equivalence. One has to find a homotopy inverse and the only thing I see, via doing this for $n = 2$, is mapping the triangle in that case to one small triangle in the twofold subdivision. Yet I do not find a homotopy.
My ultimate goal is to test the hypotheses of corollary 2.5 in https://projecteuclid.org/download/pdf_1/euclid.bams/1183542350 for $\theta = cSd^2$.
In fact the answer is that a SHE does only need to have a homotopy inverse up to a zig-zag of natural transformations from both compositions to the identity. One needs not to do it in one take, which got me confused. Then the homotopies come quite natural.