Spherical fusion categories: A certain functor 1. Context
Let $C$ be a spherical fusion category over an algebraically closed field $k$ of characteristic zero. Denote by $Vec$ the category of finite-dimensional vector spaces.
Currently, I am reading the paper Turaev-Viro invariants as an extended TQFT by Balsam and Kirillov.
On page 2 it says:

Define the functor $C^{\boxtimes n} \rightarrow Vec$ by
$\langle V_1, ..., V_n\rangle= Hom_C(1, V_1 \otimes ... \otimes V_n)$ for any collection $V_1, ... V_n$ of objects of $C$. Note that the pivotal structure gives functorial isomorphisms $$z:\langle V_1, ..., V_n\rangle \cong \langle V_n,V_1 ...,V_{n-1}\rangle$$ such that $z^n=id$; thus, up to canonical isomorphism, the space $\langle V_1, ..., V_n\rangle$ only depends on the cyclic order of $V_1, ..., V_n$.

I seems that $C^{\boxtimes n}$ refers to the $n$-fold tensor product of the category $C$. At least Lectures on tensor categories and modular functor by Bakalov and Kirillov (which is referenced in the paper) defines on page 15 (Def.1.1.15.):

Let $C_1, C_2$ be additive categories over $k$. Their tensor product $C_1 \boxtimes C_2$ is the category with the following objects and morphisms:

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*$Obj(C_1 \boxtimes C_2)$ = finite sums of the form $\bigoplus X_i \boxtimes Y_i, X_i \in Obj(C_1), Y_i \in Obj(C_2)$.

*$Hom_{C_1 \boxtimes C_2} (\bigoplus X_i \boxtimes Y_i, \bigoplus X'_j \boxtimes Y'_j)= \bigoplus\limits_{i,j} Hom(X_i, X'_j) \otimes Hom(Y_i, Y'_j)$.


2. Questions

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*What does the expression $\bigoplus X_i \boxtimes Y_i, X_i \in Obj(C_1), Y_i \in Obj(C_2)$ stand for? That is, what does the symbol $\boxtimes$ refer to?

*If $C$ is simply a spherical fusion category, how is it additive? That is, why can you consider its tensor product as defined above?

*How does the pivotal structure give these functorial isomorphisms? I tried reading the reference given in the paper, namely chapter 5.3 (which deals with Moore-Seiberg data) of Lectures on tensor categories and modular functor by Bakalov and Kirillov. However, I couldn't find a proof of that claim.

 A: In my answer, [EGNO] refers to Etingof, Gelaki, Nikshych, Ostrik: "Tensor Categories", AMS 2015, freely available on Etingof's homepage http://www-math.mit.edu/~etingof/books.html (note the copyright).


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*If $C$ is simply a spherical fusion category, how is it additive? That is, why can you consider its tensor product as defined above?


By definition, a fusion category is a $\mathbb k$-linear abelian category and thus is, in particular, additive (see [EGNO, Definition 4.1.1]). The tensor products between the Hom spaces are therefore tensor products of $\mathbb k$-vector spaces.


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*What does the expression $\bigoplus X_i \boxtimes Y_i, X_i \in Obj(C_1), Y_i \in Obj(C_2)$ stand for? That is, what does the symbol $\boxtimes$ refer to?


For $\mathbb k$-linear abelian categories $\mathcal C$, $\mathcal D$, the $\mathbb k$-linear abelian category $\mathcal C \boxtimes \mathcal D$ denotes their Deligne tensor product. The definition is given in terms of a universal property that is similar to the tensor product of other algebraic structures.
$\mathcal C \boxtimes \mathcal D$ is the $\mathbb k$-linear abelian category that is universal for functors from the category of $\mathbb k$-linear abelian categories to the category of right exact bilinear bifunctors out of $\mathcal C \times \mathcal D$, in the following sense: There is a right exact bifunctor $F_\boxtimes: \mathcal C \times \mathcal D \to \mathcal C \boxtimes \mathcal D$ such that any other right exact bifunctor $\mathcal C \times \mathcal D \to \mathcal A$ factors uniquely (in the appropriate sense) through $F_\boxtimes$ (see https://ncatlab.org/nlab/show/Deligne+tensor+product+of+abelian+categories or [EGNO, Definition 1.11.1] for details).
Accordingly, the object $X\boxtimes Y$ is the unique object of $\mathcal C \boxtimes \mathcal D$ that is associated to $(X,Y)\in \mathcal C \times \mathcal D$, i.e. $X\boxtimes Y \equiv F_\boxtimes (X,Y)$. The $\mathbb k$-linear abelian structure allows to take sums $\bigoplus_i X_i \boxtimes Y_i$ of such objects.


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*How does the pivotal structure give these functorial isomorphisms? I tried reading the reference given in the paper, namely chapter 5.3 (which deals with Moore-Seiberg data) of Lectures on tensor categories and modular functor by Bakalov and Kirillov. However, I couldn't find a proof of that claim.


The isomorphisms $\langle V_1,\ldots, V_n \rangle \overset{\cong}{\to} \langle V_n,V_1,\ldots, V_{n-1} \rangle$ are obtained in the simplest way: Suitably employ the coevaluation and evaluation on the object $V_n$ to map a morphism $1\to V_1 \otimes\ldots\otimes V_n$ to a morphism $1\to V_n \otimes V_1 \otimes\ldots\otimes V_{n-1}$. I suggest you use graphical calculus (string diagrams) to represent what is going on algebraically: First use a coevaluation to bend the string labeled $V_n$ down, and then an evaluation to bring that string all around the other factors $V_1 \otimes\ldots\otimes V_{n-1}$ to the first position. What you get is a morphism in $\langle V_n,V_1,\ldots, V_{n-1} \rangle$. Reversing these operations is the inverse, thanks to the "snake identity" ("zig-zag moves") satisfied by the duality morphisms. Here the pivotal structure has to be used, otherwise one does not get the necessary identification between left and right duals.
