Let $z$ be a fixed non-zero complex number. Let $V$ be a vertex algebra, $W_1$, $W_2$, and $W_3$ be $V$-modules. Huang defines a $Q(z)$-intertwining map between these modules to be a linear map $F:W_1\otimes W_2\rightarrow \bar{W_3}$ ($\bar{W_3}$ is the completion of $W_3$) satisfying: $$z^{-1}\delta (\frac{x_1-x_0}{z})Y_3^*(v,x_0)F(w_{(1)}\otimes w_{(2)})=$$ $$=x_0^{-1}\delta (\frac{x_1-z}{x_0})F(Y_1^*(v,x_1)w_{(1)}\otimes w_{(2)})$$ $$-x_0^{-1}\delta (\frac{z-x_1}{-x_0})F(w_{(1)}\otimes Y_2(v,x_1)w_{(2)}),$$ for all $v\in V$, $w_{(1)}\in W_1$, and $W_{(2)}$. Huang says that this corresponds to $\mathbb{CP}^1$ with punctures $z, \infty, 0$ and the local coordinates $w$, $1/w$, $w-z$.
My first question is the following: Is the fact that $Y^*_3$ and $Y^*_1$ appear explained by the fact that the local coordinates at the punctures $z$ (that is an output) and $\infty$ (that is an input) have the wrong orientation? (Also is there some way of justifying this?)
Now, the above equation is the standard way in which the $Q(z)$-intertwining map condition is used. Rather, you should first replace $v$ by $(-x_0^2)^{L(0)}e^{-x_0L(1)}v$, and then replace $x_0$ by $x_0^{-1}$. What does these two transformations correspond to in the sphere with punctures picture?
