The images are from section $2-4$ of do Carmo's Differential Geometry of Curves and Surfaces. In the following discussion $T_p(S_1)$ is the tangent plane of $S_1$ (a regular surface) at $p$ (a point of such surface). Similarly with $T_{\varphi(p)}(S_2)$.
I do not understand what it means for $\varphi$ to be differentiable since it is defined as a function $S_1\rightarrow S_2$, where $S_1$ is not an open set of $\mathbb{R}^3$. As I understand it, the goal of proposition $2$ (the proposition following the discussion above, and which is pictured below in the post) is to define what it means for a function between two regular surfaces to be differentiable, so it makes no sense to claim that $\varphi$ is differentiable at this point of the text.
In case it is needed, proposition $2$ reads as follows:
What does it mean for $\varphi$ to be differentiable?
Best Answer
For both $S_1, S_2$ there are differentiable maps $\phi_1:\Omega_1\to S_1$ and $\phi_2:\Omega_2\to S_2$ with $\Omega_1,\Omega_2$ open sets in $\mathbb R^2$, so a differentiable map $f:\Omega_1\to\Omega_2$ can be used to define $\varphi$, this is $$\phi_2\cdot f\cdot\phi_1^{-1}.$$