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


*Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor*, [**Tensor categories**](http://dx.doi.org/10.1090/surv/205), Mathematical Surveys and Monographs 205. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2024-6/hbk). xvi, 343 p. (2015). [ZBL1365.18001](https://zbmath.org/?q=an:1365.18001).


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.



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


