I'm looking for a clarification of a construction done in Seidel's [Symplectic Floer Homology of a Dehn twist][1]: I don't get why his choice of almost complex structure on $\Sigma$ is a valid one for writing the equations for fixed points Floer homology.

Let $\phi : \Sigma \to \Sigma$ be a symplectomorphism with non-degenerated fixed points. To define $HF_*(\phi)$, among many other things, we need to equip $\Sigma$ with a family of almost complex structures $J_t$ such that $J_{t+1}=\phi^*J_t$.

In the aforementioned paper, this is done in steps and the one that I don't understand is the following:

> at page 832 the author claims that on the neighbourhood of the Lagrangian sphere $[-R,R]\times S^1$ there are standard almost complex structures $\{J_t^R\}$ that can be extended to the entire $\Sigma$ 

**The problem that I have is that I don't see why $(T_0^R)^*J_t^R=J_{t+1}^R$ on the cylinder.** For me the standard almost complex structure on $[-R,R]\times S^1$ is the (constant family) that in the canonical coordinates looks like $$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$
but in this case it's immediate to verify by hand that $$(T_0^R)^*J_t^R \neq J_{t+1}^R$$
It's very likely that I'm missing something, but I really don't know what, hence I decided to ask the question once and for all.
> what are these standard almost complex structures on the cylinder that are suitable for setting up the Floer equations in this case? 


  [1]: https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1996/0003/0006/MRL-1996-0003-0006-a010.pdf