Reading the paper Floer cohomology of Lagrangian intersecitons and pseudo-holomorphic discks 2, in page $1004$ the authors want to prove that the linearization of $\bar \partial_J$ is surjective for their choice of complex structure and exact isotopy $\phi$. \par

To do this they take a $\xi \in T_{u}\mathcal{P}$, where $u\in \mathcal{M}_{J,\phi}$, and claim that $E_u(\xi)=\nabla_{s}|_{s=0} \bar \partial _J(u_s)$. What does this actually mean , since $\bar \partial _J(u_s)$ is a family of vector fields $\xi_s$ along $u_s$.

Also at some point they claim $\nabla_{s}\frac{\partial u_s}{\partial \tau}|_{s=0}= \nabla_s\frac{\partial u}{\partial \tau }|_{s=0} = \nabla_{\tau}\frac{\partial u}{\partial s}|_{s=0}$, where I don't know why we have the first inequality.

Any help or just poiting a reference where I could check this would be appreciated. Thanks in advance.

Reading the paper Floer cohomology of Lagrangian intersecitons and pseudo-holomorphic discks 2, in page $1004$ the authors want to prove that the linearization of $\bar \partial_J$ is surjective for their choice of complex structure and exact isotopy $\phi$. \par

To do this they take a $\xi \in T_{u}\mathcal{P}$, where $u\in \mathcal{M}_{J,\phi}$, and claim that $E_u(\xi)=\nabla_{s}|_{s=0} \bar \partial _J(u_s)$. What does this actually mean , since $\bar \partial _J(u_s)$ is a family of vector fields $\xi_s$ along $u_s$.

Also at some point they claim $\nabla_{s}\frac{\partial u_s}{\partial \tau}|_{s=0}= \nabla_s\frac{\partial u}{\partial \tau }|_{s=0} = \nabla_{\tau}\frac{\partial u}{\partial s}|_{s=0}$, where I don't know why we have the first inequality.

The notation is also the notation from https://stacks.stanford.edu/file/druid:bz202yk0512/immersed-augmented.pdf of page $37$.

Any help or just poiting a reference where I could check this would be appreciated. Thanks in advance.