Why is the covariant derivative of a one-form $\nabla_{i}v_j=\frac{\partial v_j}{\partial u^{i}}-\Gamma^k_{~ij}v_k$? I understood that the covariant derivative of a vector field is
$$
\nabla_{i}v^j=\frac{\partial v^j}{\partial u^{i}}+\Gamma^j_{~ik}v^k
$$
Then why is the covariant derivative of a covector field
$$
\nabla_{i}v_j=\frac{\partial v_j}{\partial u^{i}}-\Gamma^k_{~ij}v_k
$$
I tried from the first formula by lowering indices, but I just do not get the minus sign before the $\Gamma$.
Probably a trivial question of some of you, but not for a beginner of tensor calculus like myself!
 A: I follow Carroll's method of deriving this. So, we postulate that the covariant derivative is the partial derivative plus some correction (see Wald for a proof). We then have the following $$\nabla_\mu V^\nu=\partial_\mu V^\nu+\Gamma^\nu_{\mu\lambda}V^\lambda$$
Using similar reasoning to how we got the above, we postulate the following: $$\nabla_\mu V_\nu=\partial_\mu V_\nu+\gamma^\lambda_{\mu\nu}V_\lambda$$
However, we as of yet have no justification on equating $\gamma$ and $\Gamma$ just yet. They transform the same (which I will not prove here), but that's all. To prove a relation between the two, we assume two more things about the covariant derivative in addition to linearity and the Leibniz product rule: that the covariant derivative of the Kronecker delta vanishes $\nabla_\mu \delta^\nu_\lambda=0$ and that it reduces to the partial derivative on scalars $\nabla_\mu\phi=\partial_\mu\phi$. Using these, let's calculate the following: $\nabla_\mu(V_\sigma W^\sigma)$
$$\begin{align}
\nabla_\mu(V_\sigma W^\sigma)&=\nabla_\mu(\delta_\lambda^\sigma V_\sigma W^\lambda) \\
&=V_\sigma W^\lambda\nabla_\mu\delta^\sigma_\lambda+V_\sigma\nabla_\mu W^\sigma+W^\sigma\nabla_\mu V_\sigma \\
&=V_\sigma\partial_\mu W^\sigma+W^\sigma\partial_\mu V_\sigma+V_\sigma\Gamma^\sigma_{\mu\lambda}W^\lambda+W^\sigma\gamma^\lambda_{\mu\sigma}V_\lambda 
\end{align}$$
But, $V_\sigma W^\sigma$ is a scalar! Thus, by our second new property, it should reduce to the partial derivative. So, we also have $$
\begin{align}
\nabla_\mu(V_\sigma W^\sigma)&=\partial_\mu(V_\sigma W^\sigma)\\
&=V_\sigma\partial_\mu W^\sigma+W^\sigma\partial_\mu V_\sigma
\end{align}$$
Equating terms, we immediately see that $$0=\Gamma^\sigma_{\mu\lambda}V_\sigma W^\lambda+\gamma^\lambda_{\mu\sigma}W^\sigma V_\lambda$$
Relabeling indices, we see that $$\gamma^\lambda_{\mu\sigma}W^\sigma V_\lambda=-\Gamma^\lambda_{\mu\sigma}W^\sigma V_\lambda$$
Since $V_\sigma$ and $W^\sigma$ are completely arbitrary, we have in general that
$$\bbox[5px,border:1.5px solid black]
{
\gamma^\lambda_{\mu\sigma}=-\Gamma^\lambda_{\mu\sigma}
}$$
So, after much ado, we finally have, for the covariant derivative of a one-form,
$$\nabla_\mu V_\nu=\partial_\mu V_\nu-\Gamma^\lambda_{\mu\nu}V_\lambda$$
A: The covariant derivative is at first a map $\nabla:\Gamma(TM)\times\Gamma(TM)\to\Gamma(TM)$ taking two vector fields $(X,Y)$ to a third $\nabla_X Y$ obeying some properties. In particular it has to be a tensor in the first entry $\nabla_{fX}Y=f\nabla_XY$ and must be a derivation in the second entry $\nabla_{X}(fY)=X(f)Y+f\nabla_X Y$.
Then we want to actually have maps $\nabla : \Gamma(TM)\times\Gamma(T^r_sM)\to\Gamma(T^r_s M)$ which take now a vector field and a tensor field $(X,T)$ and give you a tensor field $\nabla_X T$. These maps must clearly be built from the initial one you had for vector fields only.
Now all tensor fields with $r$ indices up and no index down, i.e., elements of $\Gamma(T^r_0M)$ can be generated from vector fields by the tensor product. In that case if you demand that $$\nabla_X(T\otimes S)=\nabla_X T\otimes S+T\otimes \nabla_X S,\tag{1}$$
then the initial map, which acts only on vector fields, already fixes all the maps $$\nabla:\Gamma(TM)\times \Gamma(T^r_0M)\to \Gamma(T^r_0M).\tag{2}$$
Moreover property (1) is just again saying that $\nabla$ acts as a derivative.
Now we also have all the tensor fields with $s$ indices down and no index up, i.e., elements of $\Gamma(T^0_sM)$. All of these can be generated from the one-forms $\Gamma(T^0_1M)$ and tensor products. So if we fix $\nabla$ acting on forms, property (1) again fixes it once and for all in all $\Gamma(T^0_sM)$.
The way we go here is by demanding that if $\omega\in \Gamma(T^0_1M)$ is a one-form and $Y\in \Gamma(TM)$ is a vector field, we have $$\nabla_{X}[\omega(Y)]=(\nabla_X \omega)(Y)+\omega(\nabla_X Y)\tag{3}.$$
This is again a Liebnitz rule of sorts, where we view the contraction $\omega(Y)$ as a kind of product. Observe from (3) that since $\nabla_X Y$ is already defined, and since $\omega(Y)$ is just a smooth function, if we say how $\nabla$ acts on smooth functions we are done, this formula fully defines $\nabla_X \omega$.
Now the most natural definition is to take $\nabla_X f = X(f)$ for any smooth function. In particular, comparing the initial definition of $\nabla$ on vector fields which demanded $\nabla_X(fY)=X(f)Y+f\nabla_X Y$ and property (1), we see that these are fully compatible once we identify that $f\otimes Y = fY$.
Once this is done we have that $$\nabla_X\omega(Y) = X(\omega(Y))-\omega(\nabla_X Y),\tag{4}$$
and as I said from (1) this extends to all $T^0_sM$. Clearly now everything also extends to all $T^r_sM$.
Now to compare. Introduce a coordinate system $x^\mu$ with coordinate basis $\partial_\mu$. Denote $\nabla_\mu := \nabla_{\partial_\mu}$. Then if $X$ is a vector field $X = X^\nu\partial_\nu$ we have $$\nabla_\mu X = \nabla_\mu (X^\nu \partial_\nu)=\partial_\mu X^\nu + X^\nu \nabla_\mu \partial_\nu=\partial_\mu X^\nu \partial_\nu + X^\nu\Gamma_{\mu\nu}^\lambda \partial_\lambda=(\partial_\mu X^\lambda + \Gamma_{\mu\nu}^\lambda X^\nu)\partial_\lambda\tag{5}.$$
Where I just have used the rules for $\nabla$ and defined $\Gamma_{\mu\nu}^\lambda$ by means of $\nabla_\mu \partial_\nu = \Gamma_{\mu\nu}^\lambda \partial_\lambda$.
Finally using (4) we can evaluate the components of the one-form $\nabla_\mu \omega$ in the coordinate basis $dx^\nu$ by applying it to $\partial_\nu$. Recalling that $\omega(\partial_\nu)=\omega_\nu$ we get $$\nabla_\mu\omega(\partial_\nu)=\partial_\mu \omega_\nu-\omega(\nabla_\mu \partial_\nu)=\partial_\mu\omega_\nu-\omega(\Gamma_{\mu\nu}^\lambda\partial_\lambda)=\partial_\mu\omega_\nu-\Gamma_{\mu\nu}^{\lambda}\omega_\lambda\tag{6}$$
as you suggested.
Finally all this would work for any covariant derivative $\nabla$. It is only when you impose vanishing torsion and metric compatibility that you fix uniquely the Levi-Civita one and you are able to determine, in local coordinates, $\Gamma_{\mu\nu}^\lambda$ in terms of the metric.
So in summary it all boils down in how to define a covariant derivative. In particular in how we first define it on vectors and extends to tensors in the simples and most natural way.
