I am reeding the book by Aubin on Differential Geometry.
Let $D_XY$ be the covariant derivative of the vector field $Y$ in the direction of the vector $X$.
We know that $D_XY = X^iD_iY=X^i(\partial_i Y^j)\frac{\partial}{\partial x^j} + X^iY^jD_i\left(\frac{\partial}{\partial x^j}\right)$ where $D_i = D_{\frac{\partial}{\partial x^i}}$.
Then the author defines: $\nabla Y$ is the differential (1,1)-tensor which in a local chart has $(D_iY)^j$ as components (the $j$th component of the vector field $D_iY$.) The above implies that $\nabla_i Y^j= (D_iY)^j$.
1) This equality follows by definition if I am right. the book states it a bit confusingly.
2) What is the $\nabla$ thing called or mean? i can't find it in any book. What is the use of it?