This question is related to Pivotal functors of that are substantially different from finite group homomorphisms.
A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes called "surjective") if for any $Y: \mathcal{D}$, there is an $X: \mathcal{C}$ such that $Y$ is a subobject of $FX$.
It is known ("On fusion categories" by Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik, proposition 8.8), that for any dominant tensor functor of fusion categories $F: \mathcal{C} \to \mathcal{D}$ we have:
$$FR_\mathcal{C} = \frac{\operatorname{FPDim}(\mathcal{C})}{\operatorname{FPDim}(\mathcal{D})}R_\mathcal{D}$$ Here, $\operatorname{FPDim}$ is the Frobenius-Perron-dimension and $R_\mathcal{C} = \bigoplus \operatorname{FPDim}(X) X$ is the regular representation with respect to the Frobenius-Perron-dimensions. (The sum ranges over all simple objects.)
Now, if $\mathcal{C}$ and $\mathcal{D}$ are spherical, and $F$ is pivotal, we can ask the same for categorical dimensions. Denoting the categorical dimension by $\dim$ and defining $\Omega_\mathcal{C} := \bigoplus \dim(X) X$, we can ask:
For which pivotal, dominant tensor functors does the following equation hold? $$F\Omega_\mathcal{C} = \frac{\dim\left(\Omega_\mathcal{C}\right)}{\dim\left(\Omega_\mathcal{D}\right)}\Omega_\mathcal{D}$$ One instance where I know that it holds, is when Frobenius-Perron-dimensions and categorical dimensions coincide, i.e. in "pseudo-unitary" fusion categories, so for example (quantum) group representations. Does this hold in general? Is there a counterexample?
P.S.: Unitary fusion categories are fusion categories with a compatible C-* structure (dagger structure with Hilbert space enrichment). Pseudo-unitary fusion categories are fusion categories where $\operatorname{FPdim}(\mathcal{C}) = \dim\left(\Omega_\mathcal{C}\right)$. Unitary fusion categories are always pseudo-unitary.
P.P.S.: Here is a sub-question, that may make the original question easier to answer: What are examples for non-unitary spherical categories?
