Coderivative of symmetric $2$-tensor

Question

This is taken from a paper by M. Berger:

Here $(X,g)$ is a Riemannian manifold and $h$ is a symmetric $2$-tensor on $X$. Could you help me to understand this definition, please? What does $\nabla^k$ mean?

Is it correct to say that if $\{ E_1, \dots , E_n\}$ is a local orthonormal frame, then

$$(\delta h)_i = - \sum_k E_k(h(E_k, E_i)) \quad ?$$