I am struggling to understand why in the following highlighted part of the second image (the previous image showing how the notion of a connection is introduced in my notes on Riemannian Geomety) : 
we have that $\nabla_{e_i}(e_j)$ makes sense. Namely in the previous page (first image), $\nabla$ takes two vector fields and returns a third, but these vector fields are globally defined, whereas our local frame is a collection of vector fields on some open subset $U \subset \mathcal{M}$. I am guessing that there must be a natural operator $\nabla \restriction U$ that takes vector fields defined on $U$ and returns vector fields on $U$, and this operator must be such that it coincides with $\nabla$ when its inputs are globally defined vector fields restricted to $U$. I also suppose that we could show such an operator exists and is unique via using bump functions, but I am not sure how this would work for any random open $U \subset \mathcal{M}$.
Thanks in advance for any help.