Problem
Problem 5.36 Given a linear connection $\nabla$ on the $C^{\infty}$ manifold $M$, one defines the conjugate or opposite connection $\widehat{\nabla}$ on $M$ by
$$
\widehat{\nabla}_{X} Y=\nabla_{Y} X+[X, Y], \quad X, Y \in \mathfrak{X}(M)
$$
(i) Prove that $\widehat{\nabla}$ is a linear connection.
(ii) Compute the local components $\widehat{\Gamma}_{j h}^{i}$ of $\widehat{\nabla}$ in terms of the components of $\nabla$.