Can you extract the coefficients of particular partial derivatives from the expression below?

Questions
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Coefficient, apparently, does not descend into the sides of Equal, as in Coefficient[2 x == 3 x + 1, x]. Try something like

Map[Coefficient[#, Derivative[1, 1][f][y0, z]] &, e]
Coefficient[e /. Equal -> Subtract, Derivative[1, 1][f][y0, z]]