This question is related to this golfing question (but I'm not interested in golfing, only in functional operator composition):

How can we convert f[a,b][c,d] to {a+c,b+d} using only operator forms and function compositions?

So far I've figured out how to convert f[a][b] to a+b:

Apply[Curry[Plus]] ~ Operate ~ f[a][b]
(*    a + b    *)

but I'm stumped by the double-argument forms of the first problem, which interferes with currying. Do you know how to solve this?

This question is related to this golfing question (but I'm not interested in golfing, only in functional operator composition):

How can we convert f[a,b][c,d] to {a+c,b+d} using only operator forms and function compositions?

So far I've figured out two ways of converting f[a][b] to a+b:

Apply[Curry[Plus]] ~ Operate ~ f[a][b]
(*    a + b    *)

Plus @@ Apply[Curry[List]] ~ Operate ~ f[a][b]
(*    a + b    *)

but I'm stumped by the double-argument forms of the first problem, which interferes with currying. Do you know how to solve this?