In [FSS12, Proposition 2.10], it is demonstrated that for a finite-dimensional ribbon factorizable (so unimodular) Hopf algebra $ H $, the dual $ H^* $ possesses a Frobenius algebra structure within the category $\mathrm{Bimod}(H)$, although it has a different comultiplication and counit.

According to [EGNO15, Corollary 7.20.4], if $\mathcal{C}$ is a unimodular tensor category, then $\underline{\mathrm{Hom}}(\mathbf{1},\mathbf{1})$ in $\mathcal{C} \boxtimes \mathcal{C}^{\mathrm{op}}$ is identified as its **canonical Frobenius algebra** and is connected, as demonstrated below. This framework seems to align with [FSS12, Proposition 2.10] when $\mathcal{C} = \mathrm{Rep}(H)$, where $ H $ is a finite-dimensional unimodular Hopf algebra. As detailed in [EGNO15, Example 7.20.6], $ H^* $ serves as the canonical Frobenius algebra in the category $\mathrm{Rep}(H \otimes H^{\mathrm{cop}})$, albeit featuring a distinct comultiplication and counit.

In [LW23, Section 6], various rigid Frobenius algebras within non-semisimple modular tensor categories are explored. Recall from [LW23, Definition 3.8] that an algebra $ A $ in $\mathcal{C}$ is termed *rigid Frobenius* if it is connected, commutative, and special Frobenius. 

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**Canonical Frobenius algebra and connectedness**

In [EGNO15, Corollary 7.20.4], the Frobenius algebra $\underline{\mathrm{Hom}}(\mathbf{1},\mathbf{1})$ in $\mathcal{C} \boxtimes \mathcal{C}^{\mathrm{op}}$ is discussed for a unimodular *multitensor* category $\mathcal{C}$. However, as noted with Mainak, we have the following:

**Theorem:** The Frobenius algebra $\underline{\mathrm{Hom}}(\mathbf{1},\mathbf{1})$ is connected if and only if $\mathcal{C}$ is tensor.

*Proof:* Consider a multitensor category $\mathcal{D}$ and a $\mathcal{D}$-module category $\mathcal{M}$, with objects $M_1$ and $M_2$ in $\mathcal{M}$. According to [EGNO15, (7.20)], $\underline{\mathrm{Hom}}(M_1, M_2)$ is defined via the natural isomorphism:
$$ \mathrm{Hom}_{\mathcal{M}}(X \otimes M_1, M_2) \simeq \mathrm{Hom}_{\mathcal{D}}(X, \underline{\mathrm{Hom}}(M_1, M_2)). $$

Assume $\mathcal{M}$ is also monoidal with unit $\mathbf{1}_{\mathcal{M}}$, and let $\mathbf{1}_{\mathcal{D}}$ be the unit of $\mathcal{D}$. Set $M_1 = M_2 = \mathbf{1}_{\mathcal{M}}$ and $X = \mathbf{1}_{\mathcal{D}}$, yielding:
$$ \mathrm{Hom}_{\mathcal{M}}(\mathbf{1}_{\mathcal{M}}, \mathbf{1}_{\mathcal{M}}) \simeq \mathrm{Hom}_{\mathcal{D}}(\mathbf{1}_{\mathcal{D}}, \underline{\mathrm{Hom}}(\mathbf{1}_{\mathcal{M}}, \mathbf{1}_{\mathcal{M}})). $$

Thus, $\underline{\mathrm{Hom}}(\mathbf{1}_{\mathcal{M}}, \mathbf{1}_{\mathcal{M}})$ is connected if and only if $\mathbf{1}_{\mathcal{M}}$ is linear-simple, meaning $\mathrm{Hom}_{\mathcal{M}}(\mathbf{1}_{\mathcal{M}}, \mathbf{1}_{\mathcal{M}})$ is one-dimensional. Therefore, for a multitensor category $\mathcal{M}$, $\underline{\mathrm{Hom}}(\mathbf{1}_{\mathcal{M}}, \mathbf{1}_{\mathcal{M}})$ is connected if and only if $\mathcal{M}$ is a tensor category. The theorem follows by setting $\mathcal{D} = \mathcal{C} \boxtimes \mathcal{C}^{\mathrm{op}}$ and $\mathcal{M} = \mathcal{C}$. $\square$
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*References*

[EGNO15] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. *Tensor Categories*. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.

[FSS12] Fuchs, Jürgen; Schweigert, Christoph; Stigner, Carl. Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms. J. Algebra 363 (2012), 29--72. 

[LW23] Laugwitz, Robert; Walton, Chelsea. *Constructing non-semisimple modular categories with local modules.* Comm. Math. Phys. 403 (2023), no. 3, 1363--1409.