Let $(M,J)$ be a complex manifold with Hermitian metric $h$, and let $\nabla$ be the Chern connection on $TM^{\mathbb{C}}$, then $\nabla h = 0$ and $\nabla^{0,1}=\bar{\partial}$.
I want to show that $\nabla J = 0$. For any $X,Z\in TM^{\mathbb{C}}$, we have $$(\nabla_Z J)(X) = \nabla_Z(JX) - J\nabla_Z X,$$ so it suffices to show $\nabla_Z(JX) = J\nabla_Z X$, but I don't know how to proceed.