Questions about backward function in gspmm and gsddmm in dgl paper

I’m very confused about the gradient formulation of gsdmm and gspmm.Take sddmm as example, here is the definition of sddmm, X, Y, W are src node feat, dst node feat and edge feat:

First, the gradient \frac{\partial L}{\partial W } is a gsddmm, the Proof is a little brief and difficult to me to understand, Why this proof can prove Lemma 1 by giving gradient \frac{\partial L}{\partial w_e } . And why the \phi ' _w is \phi ' _w: R^{|V| * d_1}, R^{|V| * d_2}, R^{ |\varepsilon| * (d_3 + d4)} \mapsto R ^ {|\varepsilon| * d_4} while \phi _w is \phi _w: R^{|V| * d_1}, R^{|V| * d_2}, R^{ |\varepsilon| * (d_3)} \mapsto R ^ {|\varepsilon| * d_4} , why the third dim of \phi ' _w is (d_3 + d_4)

I would appreciate it if someone could give me a little advice.

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