Covariant Derivative: 2nd Diff - My Question

[Kisok]

TL;DR

On p158 of General Relativity written by Hobson et al., there is second differentiation of the covariant derivative. However hard I tried to understand why the first equality for the second differentiation holds, I could not. I would be very appreciated if someone help me why it should be.

My question is shown in Summary section. Please see the attached file.

 

[martinbn]

Well, what is this $\nabla_cT_{ba}$ equal to?