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!