Local geometry near a transverse intersection of two Riemann surfaces

Question

Suppose $C_1$ and $C_2$ are two Riemann surfaces (embedded as submanifolds in some ambient space, say, $M$) which intersect transversally at a point $p_1 \in C_1$. How does one see that near $C_1$, the surface $C_2$ appears as the fiber of $T^{*}C_1$ over $p_1$? (Edit: a reference will be quite sufficient too, and will be greatly appreciated! The context is a physics setting involving M-branes wrapping some Riemann surfaces.)