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 $X,Y$, 
$$
f(X) \otimes f(Y) \cong f(Z\_1) \oplus \cdots \oplus f(Z\_k)?
$$



A non-monoidal functor that respects fusion rules
I don’t think there’s a well-established name, but they’re called “quasi monoidal functors” in <https://arxiv.org/abs/1711.00645>


