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Here's a quote from Wald. I have highlighted what I think are the important parts.
From Wald, p. 33.
Thus, we have shown that [itex]\tilde\nabla_a  \nabla_a [/itex] defines a map of dual vectors at p (as opposed to dual vector fields defined in a neighborhood of p) to tensors of type (0, 2) at p. By property (1), this map is linear. Consequently [itex]\tilde\nabla_a  \nabla_a [/itex] defines a tensor of type (1,2) at p, which we will denote as [itex]C^c{}_{ab}[/itex]. Thus, we have shown that given any two derivative operators [itex]\tilde\nabla_a[/itex] and [itex]\nabla_a [/itex] there exists a tensor field [itex]C^c{}_{ab}[/itex] such that
[itex]\nabla_a \omega_b= \tilde\nabla_a \omega_b  C^c{}_{ab}\omega_c[/itex] (3.1.7)
...snip...
Continuing in a similar manner, we can derive the general formula for the action of [itex]\nabla_a[/itex] on an arbitrary tensor field in terms of [itex]\tilde\nabla_a[/itex] and [itex]C^c{}_{ab}[/itex]. For [itex]T \in {\cal T}(k,l)[/itex] we find
[itex]\nabla_a T^{b_1 \cdots b_k}{}_{c_1 \cdots c_l}=
\tilde\nabla_a T^{b_1 \cdots b_k}{}_{c_1 \cdots c_l} +
\sum_i C^{b_i}{}_{ad} T^{b_1 \cdots d \cdots b_k}{}_{c_1 \cdots c_l}
\sum_j C^{d}{}_{ac_j} T^{b_1 \cdots b_k}{}_{c_1 \cdots d \cdots c_l}
[/itex] (3.1.14)
Thus, the difference between the two derivative operators [itex]\textcolor{red}{\nabla_a}[/itex] and [itex]\textcolor{red}{\tilde\nabla_a}[/itex] is completely characterized by the tensor field [itex]\textcolor{red}{C^c{}_{ab}}[/itex].

From Wald, p. 34.
The most important application of equation (3.1.14) arises from the case where [itex]\tilde\nabla_a [/itex] is an ordinary derivative operator [itex]\partial_a [/itex]. In this case, the tensor field [itex]C^c{}_{ab}[/itex] is denoted [itex]\Gamma^c{}_{ab}[/itex] and called a Christoffel symbol. Thus, for example, we write
[itex]\nabla_a t^b = \partial_a t^b + \Gamma^b{}_{ac} t^c[/itex] (3.1.15)
Since we know how to compute the ordinary derivative associated with a given coordinate system, equation (3.1.15) (and, more generally, eq. [3.1.14] with [itex]\partial_a[/itex] and [itex]\Gamma^b{}_{ac} [/itex] replacing [itex]\tilde\nabla_a[/itex] and [itex]C^b{}_{ac}[/itex]) tells us how to compute the derivative [itex]\nabla_a[/itex] if we know [itex]\Gamma^b{}_{ac}[/itex]. Note that, as defined here, a Christoffel symbol is a tensor field associated with the derivative operator [itex]\nabla_a[/itex] and the coordinate system used to define [itex]\textcolor{red}{\partial_a}[/itex]. However, if we change coordinates, we also change our ordinary derivative operator from [itex]\partial_a[/itex] to [itex]\partial'_a[/itex] and thus we change our tensor [itex]\Gamma^c{}_{ab}[/itex], to a new tensor [itex]\Gamma'^c{}_{ab}[/itex]. Hence the coordinate components of [itex]\textcolor{red}{\Gamma^c{}_{ab}}[/itex], in the unprimed coordinates will not be related to the components of [itex]\textcolor{red}{\Gamma'^c{}_{ab}}[/itex] in the primed coordinates by the tensor transformation law, equation (2.3.8), since we change tensors as well as coordinates.

