Vector fields acting on functions arguments switching (confusion)

Question

This is from example 4.22 in "Introduction to Riemannian Manifolds" by John Lee. The set-up is that $\nabla$ is the total covariant derivative. The previous proposition is that for any tensor field $F$, $\nabla^2_{X,Y} F=\nabla_X(\nabla_Y F)-\nabla_{\nabla_X Y}F$ and I asked a question yesterday about verifying this formula for $F=f\in C^\infty(M)$ (ie a $(0,0)$-type tensor). The next example however has me very confused. The author claims that for any $u\in C^\infty(M)$, we have $$ \nabla^2(Y,X)=\nabla^2_{X,Y} u=\nabla_X(\nabla_Y u)-\nabla_{\nabla_X Y}u=Y(Xu)-(\nabla_Y X) u $$ I do not understand how he concludes the last inequality. It seems to me that he switched the arguments of $X$ and $Y$. Don't we have $\nabla_X (\nabla_Y u)=\nabla_X (Yu)=X(Yu)$?

Answer 1

This is a typo and is listed online. I believe there is also a typo in the calculation below (finding the coordinate representation of $\nabla^2 u$) as he uses this formula.

EDIT: The coordinate representation of $\nabla^2 u$ is correct as was pointed out.