I have two semisimple tensor categories $C$ and $D$. Let $K(C)$ and $K(D)$ be their Grothendieck groups. I have a specific ring homomorphism $f \colon K(C) \to K(D)$ that I would like to realize as coming from a monoidal functor $F \colon C \to D$. What I know is that for every simple object $X \in C$, $f([X])$ is the genuine class of an object in $D$ (not just a virtual object).

The tensor products on both $C$ and $D$ have a symmetry, so $K(C)$ and $K(D)$ are commutative rings. I do know that there is no monoidal functor that preserves the symmetry, but I don't care about preserving the symmetry.

Can this always be done or is there some kind of obstruction I need to consider?

Edit: Since there are many adjectives you can place in front of monoidal functor, I would like the version where $F(X \otimes Y) \cong F(X) \otimes F(Y)$ (satisfying the appropriate axioms).

I have two $\mathbb{C}$-linear semisimple tensor categories $C$ and $D$. Let $K(C)$ and $K(D)$ be their Grothendieck groups. I have a specific ring homomorphism $f \colon K(C) \to K(D)$ that I would like to realize as coming from a monoidal functor $F \colon C \to D$. What I know is that for every simple object $X \in C$, $f([X])$ is the genuine class of an object in $D$ (not just a virtual object).

The tensor products on both $C$ and $D$ have a symmetry, so $K(C)$ and $K(D)$ are commutative rings. I do know that there is no monoidal functor that preserves the symmetry, but I don't care about preserving the symmetry.

Can this always be done or is there some kind of obstruction I need to consider?

Edit: Since there are many adjectives you can place in front of monoidal functor, I would like the version where $F(X \otimes Y) \cong F(X) \otimes F(Y)$ (satisfying the appropriate axioms).

Edit 2: In light of the Simon Henry's comment, let me add that both categories are linear over the complex numbers and that the endomorphisms of a simple object are just scalar multiples of the identity.

I have two semisimple tensor categories $C$ and $D$. Let $K(C)$ and $K(D)$ be their Grothendieck groups. I have a specific ring homomorphism $f \colon K(C) \to K(D)$ that I would like to realize as coming from a monoidal functor $F \colon C \to D$. What I know is that for every simple object $X \in C$, $f([X])$ is the genuine class of an object in $D$ (not just a virtual object).

The tensor products on both $C$ and $D$ have a symmetry, so $K(C)$ and $K(D)$ are commutative rings. I do know that there is no monoidal functor that preserves the symmetry, but I don't care about preserving the symmetry.

Can this always be done or is there some kind of obstruction I need to consider?

I have two semisimple tensor categories $C$ and $D$. Let $K(C)$ and $K(D)$ be their Grothendieck groups. I have a specific ring homomorphism $f \colon K(C) \to K(D)$ that I would like to realize as coming from a monoidal functor $F \colon C \to D$. What I know is that for every simple object $X \in C$, $f([X])$ is the genuine class of an object in $D$ (not just a virtual object).

The tensor products on both $C$ and $D$ have a symmetry, so $K(C)$ and $K(D)$ are commutative rings. I do know that there is no monoidal functor that preserves the symmetry, but I don't care about preserving the symmetry.

Can this always be done or is there some kind of obstruction I need to consider?

Edit: Since there are many adjectives you can place in front of monoidal functor, I would like the version where $F(X \otimes Y) \cong F(X) \otimes F(Y)$ (satisfying the appropriate axioms).