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 $X,Y$, $$ f(X \otimes Y) \cong f(Z_1) \oplus \cdots \oplus f(Z_k)? $$

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 $X,Y$, $$ f(X) \otimes f(Y) \cong f(Z_1) \oplus \cdots \oplus f(Z_k)? $$