In chapter 3 of Wald's General Relativity he starts by defining a covariant derivative $\nabla$ as a map on a manifold M from tensor fields $\mathscr{T}(k,l) \to \mathscr{T}(k,l+1)$ plus some required properties (linearity, Leibniz rule, etc.).
He then goes on to show that for any two derivatives $\nabla, \tilde{\nabla}$, their difference (applied to a one-form) can be expressed by a tensor as $$ \nabla_a \omega_b - \tilde{\nabla}_a \omega_b = C^c_{ab} \omega_c. $$ What I don't understand is that he says we choose $\tilde{\nabla}$ as the usual partial derivative $\partial$ and call the tensors $C^c_{ab} = \Gamma^c_{ab} $ the Christoffel symbols. I thought the partial derivative does not satisfy the required transformation properties of the covariant derivative hence I can't substitute it for $\tilde{\nabla}$.
Another minor issue is that he calls $C^c_{ab}$ a tensor field while he also says it doesn't transform according to the tensor transformation law. What does he then mean by that? That it is a multilinear map?