Covariant derivative of a covariant tensor
= Covariant derivative of a covariant tensor
For $W_{\mu\nu}$,
$$
\nabla_\alpha W_{\mu\nu}
=\partial_\alpha W_{\mu\nu}
-\Gamma^\rho_{\alpha\mu}W_{\rho\nu}
-\Gamma^\rho_{\alpha\nu}W_{\mu\rho}.
$$
= Covariant derivative of a covariant tensor
For $W_{\mu\nu}$,
$$
\nabla_\alpha W_{\mu\nu}
=\partial_\alpha W_{\mu\nu}
-\Gamma^\rho_{\alpha\mu}W_{\rho\nu}
-\Gamma^\rho_{\alpha\nu}W_{\mu\rho}.
$$