Coefficient, apparently, does not descend into the sides of Equal, as in Coefficient[2 x == 3 x + 1, x]. Try something like
Coefficient
Equal
Coefficient[2 x == 3 x + 1, x]
Map[Coefficient[#, Derivative[1, 1][f][y0, z]] &, e] Coefficient[e /. Equal -> Subtract, Derivative[1, 1][f][y0, z]]
