Name for this axiom $(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$

Question

I am trying to give a name to this axiom in a definition:

$(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$

(for all $X, Y, R, S$) where $\sqcup$ is the join of a lattice and $\bullet$ is some binary operation. It feels related to monotonicity/distributivity but I don't know a standard name for this. Any ideas? So far I have called it "full distributivity". I'd also like to have a name (possibly the same) for this axiom when $\sqcup$ isn't a lattice operation, just some binary operation.

Answer 1

In higher category theory, something very similar is called the interchange law, and is an axiom for a 2-category and similar structures. It's also one of the preconditions for the Eckmann-Hilton argument, where I've also occasionally seen it called the interchange property.

Answer 2

Of course this makes sense for arbitrary two binary operation. It says that $\sqcup$ and $\bullet$ commute with each other. Equivalently, $\bullet$ is a $\sqcup$-homomorphism (and by symmetry, also $\sqcup$ is a $\bullet$-homomorphism).