What is this property to be called?

Two functions, $f$ and $g$, that satisfy the following identity: $$f(g(a_1,...,a_n), g(b_1,...,b_n),...) = g(f(a_1,b_1,....), f(a_2,b_2,...)...)$$ (notice the "transposition" of the arguments), do they possess a named and/or known property? Or as an alternative, where can I find out more about it?

(Apologies if I asked a duplicate question, but I didn't know how to search for it)

• Maybe it would be helpful to write it $f\left(g(A^1),g(A^2),\ldots ,g(A^n)\right)=g\left(f(A_1),f(A_2),\ldots , f(A_n)\right)$ – Alexander Gruber Jul 17 '18 at 0:16

Wikipedia calls it "$f$ and $g$ commute" for multivariate functions:
The notion of commutation also finds an interesting generalization in the multivariate case; a function $f$ of arity $n$ is said to commute with a function $g$ of arity $m$ if $f$ is a homomorphism preserving $g$, and vice versa i.e.: $${f\big(g(a_{11},\ldots ,a_{1m}),\ldots ,g(a_{n1},\ldots ,a_{nm})\big)=g\big(f(a_{11},\ldots ,a_{n1}),\ldots ,f(a_{1m},\ldots ,a_{nm})\big)}$$