Tangent Spaces of Parametrized Sets - McInerney, Defn 3.3.5

[Math Amateur]

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ...

I am currently focussed on Chapter 3: Advanced Calculus ... and in particular I am studying Section 3.3 Geometric Sets and Subspaces of $T_p ( \mathbb{R}^n )$ ...

I need help with a basic aspect of Definition 3.3.5 ...

Definition 3.3.5 reads as follows:

In the above definition we find the following:

" ... ... Here, when we say that a parametrized curve $c \ : \ I \longrightarrow S$ is smooth, we mean that there is a smooth function $\tilde{c} \ : \ I \longrightarrow U$ such that $c = \phi \circ \tilde{c}$ ... ... "

My question is as follows:

Why do we need to bother defining $\tilde{c}$ ... the codomain of $c$ is defined as $S$ ... so we surely only need to stipulate that $c$ is continuously differentiable or $C^1$ ... that is the usual definition of 'smooth' so why isn't this enough ...

... ... so, my question is then, why do we bother defining $\tilde{c}$ and then go on to consider the composite function $c = \phi \circ \tilde{c}$ ... ?Hope someone can help ...

Peter===========================================================

To give the context for McInerney's approach to this definition I am providing the introduction to Section 3.3 as follows:

 

[fresh_42]

You can imagine the following situation. Think of $S$ being the city in which you live, $p$ your home and $U$ a roadmap of it.
The formula basically states that you can use a path $I$ on your map and project it to the actually city as well as finding it in reality.

The point is that $S$ can be of any shape, e.g. curved (or with hills in the example above).
Smoothness is a local property, i.e. it holds on small neighbourhoods around $p$, even if for any $p$. Instead to define what small neighbourhoods in $S$ are, we take a chart $(U,\phi)$ of $S$ in $ℝ^n$ where we already know what smooth means and define it with the help of $\tilde{c}$ and the requirements that $\tilde{c}$ is smooth and $c=\phi \cdot \tilde{c}$.