The Christofel symbol can be thought of as describing how a coordinate system twists and stretches as you move about on the manifold. It thus is about a particular coordinate system in two different ways:

- It describes a particular coordinate system
- and it is expressed in that same coordinate system.

It is possible to express the twisting and stretching of one coordinate system in another coordinate system. If we keep the latter coordinate system unchanged then that is a tensor expression referring to some subject coordinate system.

This is the source, I think, of the confusion in this discussion of the covariant derivative.