Statement verification - Stable and unstable manifold theorem 
Let $\dot{X} = f(X)$ have hyperbolic fixed point $\overline{X}$ and linearisation $\dot{X} = Df(\overline{X})X$. Then there exists a stable manifold $W^s_{\overline{X}}$ of dimension $d_s$ and an unstable manifold $W^u_{\overline{X}}$ of dimension $d_u$, $\color{red}{\text{containing }\overline{X} \text{ and tangent to } \overline{X}+E^s, \text{ respectively} \overline{X} + E^u \text{ at } \overline{X}.}$

I'm struggling to understand the english of the part in red... are we saying that $W^s_\overline{X}$ is a manifold containing $\overline{X}$ and is tangent to $\overline{X} + E^s$ at $\overline{X}$, and $W^u_\overline{X}$ is a manifold containing $\overline{X}$ and is tangent to $\overline{X} + E^u$ at $\overline{X}$?

Yes, that's what it means.
(I think it's a rather obtuse and needlessly confusing way to write it.)
