Expression with covariant derivative of a vector component I saw this formulation :
"If by $\nabla_{X_{i}}V_{j}$ we meant "take the covariant derivative of the components $V_{j}$ of the vector $V$", then this would simply be $X_{i}V_{j}$ as the components are just smooth functions."
with $\{e^{a}=dx^{a}\}$ basis and dual basis $\{X_{a}=\frac{\partial}{\partial x^{a}}\}$.
The expression of covariant derivative of "a-th" component of vector $V$ is :
$$
(\nabla_{X_{i}}V_{a})\,=X_{i}\,V_{a}--V_{c}\Gamma_{ia}^{c}= \,\dfrac{\partial V_{a}}{\partial x^{i}}-V_{c}\Gamma_{ia}^{c}
$$
So I would like to know why, saying "then this would simply be $X_{i}V_{j}$ as the components are just smooth functions", one gets : $\nabla_{X_{i}}V_{a}=X_{i}V_{a}$ ?? I mean, Why does term with Christoffel symbols disappear ?
I don't understand the link between smooth functions for components of vector $V$ and the vanishing of right term in expression of $\nabla_{X_{i}}V_{a}$.
Thanks
UPDATE 1
Here's a capture on the context of the formulation that confuses me :

ps : the top formulation (italic) comes from this post
 A: The expression $\nabla_{X_i} V_a$ should be read as $(\nabla_{X_i} V)_a$ and not $\nabla_{X_i} (V_a)$. Indeed, the formula 
$$ (\nabla_{X_i} V)_a = X_i V_a + \Gamma_{ia}^c V_c $$
shows that the right hand side depends not only on $V_a$ (the $a$-th coordinate of $V$ with respect to the frame $X_i$) but on all of $V_1,\dots,V_n$ (that is, all the coordinates of $V$) so you don't want to think of $\nabla_X V_a$ as an operator which eats the $a$-th coordinate of a vector field and outputs the $a$-th coordinate of the covariant derivative. 
In fact, ones sometimes denotes the directional derivative of a function $f$ in the direction of a vector field $X$ by
$$ Xf = df(X) = \nabla_X (f) $$
so using this notation, the formula above reads as
$$ (\nabla_{X_i} V)_a = \nabla_{X_i} (V_a) + \Gamma_{ia}^c V_c $$
which means that the $a$-th coordinate of the covariant derivative of $V$ in the direction $X_i$ is the directional derivative of the $a$-th component of $V$ in the direction $X_i$ plus correction terms involving the Christoffel symbols and all the other components of $V$.

The bottom line is that you shouldn't think about "the covariant derivative of the $a$-th component of a vector field" because this is not a well-defined notion without providing all the other components but rather "the $a$-th component of the covariant derivative of a vector field".
