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?