In group theory the following structure appears quite often: a sequence of monoids or groups $(G_n)_{n \geq 0}$, s.t. $G_0 = \{e\}$ together with associative and unital products $\otimes: G_k \times G_l \to G_{k+l} $ such that $$(a \otimes b) \circ (c \otimes d)=(a \circ c) \otimes (b \circ d).$$ Examples would be the set of all finite symmetric groups $S_n$ or braid monoids/groups $B_n$.
Is there a name for this kind of structure? The reason I ask is that I'm dealing with exactly this kind of structure one categorical level higher, so I have 2-groups instead of groups, etc and I don't want to invent new names for well-known structures.