Isomorphism between Davydov-Yetter complex and Hochschild complex of canonical algebra on a multitensor category

Question

I'm trying to follow the proof of proposition 7.22.7 from

Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor, Tensor categories, Mathematical Surveys and Monographs 205. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2024-6/hbk). xvi, 343 p. (2015). ZBL1365.18001.

I think I understand everything but the assignment $J$ which takes an endomorphism in $C^{\boxtimes n+1}$ of $A(n)$ to an endomorphism of $\otimes^{n}$. Meaning that for any $V_{1},\dots, V_{n}$ we need to get an endomorphism of $V_{1}\otimes\dots\otimes V_{n}$.

Here $\boxtimes$ is the Deligne tensor product, and $A(n)$ is the object in $C^{\boxtimes n+1}$ representing the functor $Hom_{C}(\otimes^{n+1}(\_),1):C^{\boxtimes n+1}\to Vect$.

If I for instance follow the example 7.22.8 in the middle of the proof, I agree that $A(n)=\bigotimes_{V_{1},\dots, V_{n}\in \mathcal{O}(C)} (V_{1}\otimes\dots\otimes V_{n})\boxtimes V_{1}^{\ast}\boxtimes\dots \boxtimes V_{n}^{\ast}$ indexed over isomorphism classes of simple objects.

But already for $n=1$ and for $C$ a fusion category as in the example, if $f\in End_{C\boxtimes C}(\bigoplus_{V\in \mathcal{O}(C)} V\boxtimes V^{\ast})$, then f is meant to be assigned, for any object $V_{1}\in C$, to the endomorphism

$id_{C}\boxtimes Hom_{C}(1,\_\otimes V_{1})(\bigoplus_{V\in \mathcal{O}(C)} V\boxtimes V^{\ast})(f)$

The authors claim in particular that $id_{C}\boxtimes Hom_{C}(1,\_\otimes V_{1})(\bigoplus_{V\in \mathcal{O}(C)} V\boxtimes V^{\ast})\cong V_{1}$ and so this assignment indeed is an endomorphism of, in this case, $V_{1}$.

However I do not see this isomorphism and in fact as $Hom_{C}(1,V^{\ast}\otimes V_{1})\in Vect$, I'm struggling to see how this object in $C\boxtimes Vect$ is being identified with one in $C$. I might be doing something silly but my calculation yields only $\bigoplus V\boxtimes Hom(1,V^{\ast}\otimes V_{1})$ which I do not see how to reduce to $V_{1}$.

Any help or reference would be appreciated.

Answer 1

The category $$\mathsf{Vect}$$ behaves like a unit with respect to the Deligne tensor $$\boxtimes$$. I think the technical way to say it is that there is a canonical 2-natural equivalence $$\mathcal{C}\boxtimes\mathsf{Vect}\simeq\mathcal{C}$$. If we use this equivalence to identify $$\mathcal{C}$$ with $$\mathcal{C}\boxtimes\mathsf{Vect}\subseteq\mathcal{C}\boxtimes\mathcal{C}$$ then the isomorphism follows from the fact that $$\mathsf{Hom}(\mathbf 1 , V^*\otimes V_i)\cong\mathsf{Hom}(V , V_i)\cong \delta_{V,V_i}\cdot k$$.

Technically I think that should be a left dual $$^*V$$, instead of the way it is written, but I will need to go back and read the proof.