Uniqueness of covariant derivative in Do Carmo 2.2 Proposition: Let $M$ be a differentiable manifold with an affine connection $\nabla$. There exists a unique correspondence which associates to a vector field $V$ along the differentiable curve $c:I\: M$ another vector field $\frac{DV}{dt}$ along $c$, called the covariant derivative of $c$ such that: 
a)$\frac{D}{dt}(V+W)=\frac{DV}{dt}+\frac{DW}{dt}$
b)$\frac{D}{dt}(fV)$=$\frac{df}{dt}V+f\frac{DV}{dt}$
c)If $V$ is induced by a vector field $Y\in \mathcal{X}(M)$, then $\frac{DV}{dt}=\nabla_{dc/dt}Y$.
Proof: Let us suppose initially that there exists such a correspondence. Let $\sum v^jX_j$ denote a vector field $V$ in local coordinates, and where $X_j=\frac{\partial}{\partial x_j}$. By a) and b) then
$$\frac{DV}{dt}=\sum_j\frac{dv^j}{dt}X_j+\sum_j v^j\frac{DX_j}{dt}$$
by c) and i)
$$\frac{DX_j}{dt}=\sum_i \frac{dx_i}{dt}\nabla_{X_i}X_j$$
Therefore, 
$$\frac{DV}{dt}=\sum_j \frac{dv^j}{dt}X_j+\sum_{i,j}\frac{dx_i}{dt}v^j\nabla_{X_i} X_j$$
The expression above shows us that if there is a correspondence satisfying the condition of proposition 2.2, then such a correspondence is unique.
So my question is why does producing the formula in local coordinates implies uniqueness. I just don't get Do Carmo's reasoning here. Doesn't he have to check compatibility conditions with respect to other charts?

From the comments above by @PaulSinclair.

If you have two operators that both satisfy conditions (a), (b), (c), then on a given coordinate system, they both have to have that same expression, and therefore must be equal within the domain of that coordinate system. Since the coordinate system itself is arbitrary, this is true for every coordinate system, which means the two operators are equal everywhere. Note that this argument starts by assuming that the operators are well-defined, so there is no need to check if they are compatible with all coordinate systems.
