target space are anti-isomorphic, and they are separable Frobenius algebras over $k$. This was first proved for weak Hopf algebras (see [@BohmNSI]), and then generalized to weak bialgebras (see [@Schauenburg]).

[\[eq:1.2a\]]{} [@Schauenburg] Let $H$ be a weak bialgebra. Then $\ol{\varepsilon}_s$ restricts to an anti-algebra isomorphism $H_t\to H_s$ with inverse $\varepsilon_t$, and $\ol{\varepsilon}_t$ restricts to an anti-algebra isomorphism $H_s\to H_t$ with inverse $\varepsilon_s$.

[\[pr:1.2b\]]{} [@Schauenburg] Let $H$ be a weak bialgebra. Then $H_s$ and $H_t$ are Frobenius separable $k$-algebras. The separability idempotents of $H_t$ and $H_s$ are $$e_t=\varepsilon_t(1_{(1)})\ot 1_{(2)}=1_{(2)}\ot \ol{\varepsilon}_t(1_{(1)});$$ $$e_s=1_{(1)}\ot \varepsilon_s(1_{(2)})=\ol{\varepsilon}_s(1_{(2)})\ot 1_{(1)}.$$ The Frobenius systems for $H_t$ and $H_s$ are respectively $(e_t,\varepsilon_{|H_t})$ and $(e_s, \varepsilon_{|H_s})$. In particular, we have for all $z\in H_t$ that $${\label{eq:1.2b.1}}
z\varepsilon_t(1_{(1)})\ot 1_{(2)}=
\varepsilon_t(1_{(1)})\ot 1_{(2)}z.$$

It was shown in [@NVT] that the category of modules over a weak Hopf algebra is monoidal; it follows from the results of [@Schauenburg] that this property can be generalized to weak bialgebras. We explain now how this can be done directly.\
Let $M$ be a left $H$-module. By restriction of scalars, $M$ is a left $H_t$-module; $M$ becomes an $H_t$-bimodule, if we define a right $H_t$-action by $$m\cdot z=\ol{\varepsilon}_s(z)m.$$\
Let $M,N\in {}_H{\mathcal{M}}$, the category of left $H$-modules. We define $$M\ot_t 