of $W$. On the other hand, because of the truncation condition any weak $V$-module $W$ is a restricted $g(V)$-module where $v(n)$ is represented by $v_{n}$ for $v\in V, \;n\in {\Z}$.

We now recall the theory of [*contragredient module*]{} established in \[FHL\]. Let $W=\coprod_{h\in {\C}}W_{(h)}$ be a $V$-module and let $W'=\coprod_{h\in {\C}}W_{(h)}^{*}$, the [*restricted dual*]{} of $W$. For $v\in V,\; w'\in W'$, define $$\begin{aligned}
\label{ey'}
\<Y'(v,x)w',w\>
=\left\<w',Y\left(e^{xL(1)}\left(-x^{-2}\right)^{L(0)}v,x^{-1}\right)w
\right\>\end{aligned}$$ for $w\in W$. Then we have the following fundamental result due to Frenkel, Huang and Lepowsky (\[FHL\], Theorem 5.2.1 and Proposition 5.3.1).

The pair $(W',Y')$ carries the structure of a $V$-module and $(W'',Y'')=(W,Y)$.

Let $W$ be a weak $V$-module for now. For $v\in V$, set (cf. \[HL1\]) $$\begin{aligned}
\label{eyovx}
Y^{o}(v,x)=Y(e^{xL(1)}(-x^{-2})^{L(0)}v,x^{-1})\in (\End\; W)[[x,x^{-1}]].\end{aligned}$$ (Note that $e^{xL(1)}(-x^{-2})^{L(0)}v\in V[x,x^{-1}]$.) For instance, $$\begin{aligned}
& &Y^{o}({\bf 1},x)=1,\label{yo1}\\
& &Y^{o}(\omega,x)=x^{-4}Y(\omega,x^{-1})\label{eyomega}\end{aligned}$$ because $L(0){\bf 1}=L(1){\bf 1}=0$, $L(0)\omega=2\omega$ and $L(1)\omega=0$, where $\omega$ is the Virasoro element of $V$. Since $e^{xL(1)}(-x^{-2})^{L(0)}v\in V[x,x^{-1}]$ for $v\in V$ and $Y(u,x^{-1})w\in W((x^{-1}))$ for any $u\in V$, we have $$\begin{aligned}
\label{eyovxw}
Y^{o}(v,x)w\in W((x^{-1}))\;\;\;\mbox{ for }w\in W.\end{aligned}$$ It was observed in \[HL1\] that FHL’s proof in \[FHL\] for Proposition \[pfhl0\] in fact proves the following [*opposite Jacobi identity*]{}: $$\begin{aligned}
\label{ehl}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y^{o}(v,x_{2})Y^{o}(u,x_{1})
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y^{o}(u,x_{1})Y^{o}(v,x_{2})\nonumber\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y^{o}(Y(u,x_{0})v,x_{2}).\end{aligned}$$ As it 