Comparing diagonals on the associahedra

We prove that the formula for the diagonal approximation $\Delta_K$ on J. Stasheff’s $n$-dimensional associahedron $K_{n+2}$ derived by the current authors in $\href{ https://dx.doi.org/10.4310/HHA.2004.v6.n1.a20}{[7]}$ agrees with the “magical formula” for the diagonal approximation $\Delta^\prime_K$ derived by Markl and Shnider in $\href{ https://www.ams.org/journals/tran/2006-358-06/S0002-9947-05-04006-7/ }{[5]}$, by J.-L. Loday in $\href{ https://doi.org/10.1007/978-0-8176-4735-3_13 }{[4]}$, and more recently by Masuda, Thomas, Tonks, and Vallette in $\href{ https://doi.org/10.5802/jep.142}{[6]}$.