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 2004 agrees with the "magical formula" for the diagonal approximation $\Delta_{K}^{\prime}$ first derived by Markl and Shnider in 2006 and more recently by Masuda, Thomas, Tonks, and Vallette in 2021.