# Does this notion related to species/operads/FI-modules have a name?

Let $B$ be the symmetric monoidal category of finite sets and bijections with disjoint union. Let $C$ be a symmetric monoidal category. Is there a standard name for a lax monoidal functor $F:B \to C$? In other words, we are considering a sequence $F(n)$ of $S_n$-representations in $C$ (i.e. a species in $C$) together with $S_n\times S_m$-equivariant maps $F(n) \otimes F(m) \to F(n+m)$ satisfying certain associativity conditions.

I could invent a name for this notion (like "a $B^\otimes$-module in $C$") but if there is already an established terminology I'd prefer to use that.

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