TITLE: Internal Hom of Deligne' tensor product
QUESTION [5 upvotes]: I read the following statement (equation 22) in "Monoidal 2-structure of bimodule categories" by Justin Greenough:
Let $\mathcal{C}$ be a finite tensor category (abelian k-linear rigid monoidal category with simple unit and finite dimensional Hom spaces). Let $\mathcal{M}$ and $\mathcal{N}$ be exact left module categories over $\mathcal{C}$. 
We introduce the left $\mathcal{C}$-module structure in $\mathcal{M} \boxtimes \mathcal{N}$ (the Deligne' tensor product of $\mathcal{M}$ and $\mathcal{N}$) by:
$$X \otimes (M \boxtimes N) = (X \otimes M) \boxtimes N,$$
where $X \in \mathcal{C}$.
Then the equation 22 tells us that
$$\underline{Hom}_{\mathcal{M} \boxtimes \mathcal{N}}(M \boxtimes N, S \boxtimes T) = \underline{Hom}_{\mathcal{M}}(M, S) \otimes \underline{Hom}_{\mathcal{N}}(N,T),$$
where $\underline{Hom}_{*}$ are internal hom for left $\mathcal{C}$ structure in
$\mathcal{M} \boxtimes \mathcal{N}$, $\mathcal{M}$ and $\mathcal{N}$.
Now, let us consider the simple case:
let $\mathcal{C}$ be a unitary fusion category and $\mathcal{M} = \mathcal{N} =  \mathcal{C}$. Then by the definition of internal Hom and the equation above, we have
$$Hom_{\mathcal{C} \boxtimes \mathcal{C}}(1 \boxtimes 1, X \boxtimes X^*) \cong Hom_{\mathcal{C}}(1, \underline{Hom}_{\mathcal{C} \boxtimes \mathcal{C}}(1 \boxtimes 1, X \boxtimes X^*))\\
\cong Hom_{\mathcal{C}}(1, \underline{Hom}_{\mathcal{C}}(1, X) \otimes \underline{Hom}_{\mathcal{C}}(1,X^*)), $$
where $1$ is the unit of $\mathcal{C}$ and $X$ is a simple object in $\mathcal{C}$ such that $X \ncong 1$ and $X^*$ is the left (or right) adjont of $X$.
Since $\underline{Hom}_{\mathcal{C}}(1, X)  = X$ and $\underline{Hom}_{\mathcal{C}}(1, X^*)  = X^*$, we have
$$\{0\} = Hom_{\mathcal{C}}(1,X) \otimes Hom_{\mathcal{C}}(1, X^*) \cong 
Hom_{\mathcal{C} \boxtimes \mathcal{C}}(1 \boxtimes 1, X \boxtimes X^*) \cong 
Hom_{\mathcal{C}}(1, X \otimes X^*) \neq \{0\}.$$
So it seems that we have a contradiction here. Can anyone tell me if I made a mistake somewhere? Does the equation (22) in "Monoidal 2-structure of bimodule categories" hold? Thank you in advance!

REPLY [4 votes]: That equation is not correct. You should be suspicious because the definition of the $\mathcal{C}$-module category structure on $\mathcal{M} \boxtimes \mathcal{N}$ doesn't use the $\mathcal{C}$-module category structure on $\mathcal{N}$, only the one on $\mathcal{M}$. So how could the $\mathcal{C}$-enriched hom of $\mathcal{N}$ show up?  
The correct equation is:
$$ \underline{Hom}_{\mathcal{M} \boxtimes \mathcal{N}}(M \boxtimes N, S \boxtimes T) \cong \underline{Hom}_{\mathcal{M}}(M , S) \otimes {Hom}_{\mathcal{N}}( N, T)$$
Here the second factor is just the usual vector space valued hom in $\mathcal{N}$ (and we are using that $\mathcal{C}$ is naturally a $Vect$-module category). 
With this corrected equation you no longer get a contradiction because it is $Hom_{\mathcal{C}}(1, X^*) = 0$ which shows up instead of $\underline{Hom}_{\mathcal{C}}(1, X^*) = X^*$.