Covariant derivative of a metric tensor

Question

I am studying differential geometry and in definition of a affine connection to be metric I have read: $$\nabla g=0$$ where $g=g_{ij}dx^i\otimes dx^j$ and $\nabla g=g_{ij;k}dx^i\otimes dx^j\otimes dx^k$. I don't understand the notation $g_{ij;k}$. I have thinked that it is a compact notation referred to $(\nabla_{\partial_k}g)(\partial_i, \partial_j)$, it is correct? Thanks in advance!