Let $(\cal{C},\otimes)$ and $(\cal{D},\odot)$ be two monoidal categories. Moreover, assume that $\cal{C}$ and $\cal{D}$ are abelian and semisimple. Let $X,Y$ be two simple objects in $\cal{C}$, and let $$ X \otimes Y \cong Z_1 \oplus \cdots \oplus Z_k, $$ be their decomposition into simple objects. Is there a name for a (not necessarily monoidal) functor $f:\cal{C} \to \cal{D}$ which satisfies, for all simple $X,Y$, $$ f(X) \otimes f(Y) \cong f(Z_1) \oplus \cdots \oplus f(Z_k)? $$

## 1 Answer

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I don’t think there’s a well-established name, but they’re called “quasi monoidal functors” in https://arxiv.org/abs/1711.00645

allobjects $A$ and $B$, but without postulating any coherences, right? $\endgroup$