Let $\mathcal C$ be a [*added later:* semi-simple] tensor category, and let $A=(A,m:A\otimes A\to A,i:1\to A)$ be an algebra object in $\mathcal C$.
The algebra is...

**Separable** if there is an $A$-$A$-bimodule map $\Delta:A\to A\otimes A$ such that $m\circ\Delta=\mathrm{id}_A$

**Frobenius** if there is an $A$-$A$-bimodule map $\Delta:A\to A\otimes A$ that is coassociative and counital.

I'm wondering whether separable implies Frobenius.

I can show that separable implies coassociative but I suspect that separable does not imply counital. I'm having a hard time finding counterexamples.

PS: by a "tensor category", I mean a category that is monoidal (not necessarily symmetric) and linear over some field $k$.

PS2: In a previous version of this question, I had added the condition that $\mathcal C$ be rigid. I'd be actually more interested to have counterexamples where the ambient category $\mathcal C$ is semi-simple. I understand that there are many things one might mean by "semisimple" (e.g., is the category of *all* vector spaces over $k$ semisimple?), I'm deliberately keeping things vague to allow for more counterexamples.