Let $k$ be a field (of characteristic zero if necessary, algebraically closed if necessary) and let $z\in k$.
Let $\phi:k[T]\rightarrow k[X,Y,...]$ be the $k$-morphism given by $T\mapsto X+Y$. Let $k[T]^h_{(T-z)}$ be the henselization of the localization at $(T-z)$ of $k[T]$. Can I identify the following two rings? $$k[T]_{(T-z)}^h\otimes k[X,Y,\frac{1}{z-Y}]\overset{?}=k[T]_{(T-z)}^h\otimes k[X,Y,\frac{1}{X}]$$
(tensor products are taken over $k[T]$ with respect to $\phi$ and the inclusion of $k[T]$ in $k[T]^h_{(T-z)}$)
(Please, if someone has a better idea for the title feel free to edit!!)
EDIT: The trick I want to use is that I (think I) can identify $k[T]_{(T-z)}^h$ with the algebraic part of $k[[T-z]]$ and I can "almost" use the formula $$\frac{-1}{Y-z}=\frac{1}{X}\cdot\frac{-1}{\frac{Y+X-z}{X}-1}=\sum_{n=0}^\infty X^{-n-1}(Y+X-z)^n$$ since the rhs can be identified with an element in $k[T]^h_{(T-z)}\otimes k[Y][[\frac{1}{X}]]$ which is "almost" $k[T]_{(T-z)}^h\otimes k[X,Y,\frac{1}{X}]$. Is there any way to adapt this "almost" argument to my case?