Confusion definition $\mathcal{F}_m(M)$ 
Let M be a $C^\infty$-manifold, then $\mathcal{F}_m(M)$  is the set of $C^\infty$-functions defined on an allowed chart $(U,\phi)$ around $m\in M$.

We defined an $C^\infty$-function as a $C^\infty$-function between the two manifolds M and $\mathbb{R}$: $f:M\rightarrow \mathbb{R}$. The set of al these functions is written as $\mathcal{F}(M)$.
What does this yellow statement exactly mean? I understand the definition of an $C^\infty$-function, but what is the difference between $\mathcal{F}(M)$ and $\mathcal{F}_m(M)$? 
 A: I believe this is the same as the stalk at $m$.  I also don't think it is necessary to mention charts in the definition, strictly speaking.
Let $U$ be an open neighborhood of $m$.  Then $\mathcal F(U)$, the set of smooth functions from $U$ to $\mathbb R$, is a ring.  If $U \subset V$ are open neighborhoods of $m$ then restriction induces a ring homomorphism $\mathcal F(V) \rightarrow \mathcal F(U)$.  Thus we get a directed system of rings, indexed by the set of open neighborhoods of $m$, partially ordered by reverse inclusion.  Then $\mathcal F_m(M)$ is defined to be the ring
$$\mathcal F_m(M) = \varinjlim\limits_U \mathcal F(U)$$
Concretely, $\mathcal F_m(M)$ can be described as follows: define an equivalence relation on the set of all pairs $[U,f]$, where $U$ is an open neighborhood of $m$ and $f: U \rightarrow \mathbb R$ is a smooth function: set $[U,f]$ to be equivalent to $[V,g]$ if there exists an open neighborhood $W$ of $m$, contained in $U \cap V$, on which $f$ and $g$ agree.  Then $\mathcal F_m(M)$ is the set of such equivalence classes.  It has a natural ring structure.  For example, if $(U,f)$ and $(V,g)$ are in $\mathcal F_m(M)$, define $(U,f) + (V,g)$ to be $$(U \cap V, f|_{U \cap V} + g|_{U\cap V})$$
