  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 $\Delta_{K}^{\prime}$ derived by Markl and Shnider in 2006, by
Loday in 2011, and by Masuda, Thomas, Tonks, and Vallette in 2021.
