The abstract tensor product and the concrete one for covariant tensors coincide, since

\begin{equation*} V_1^*\otimes \cdots \otimes V_k^* \cong L(V_1,\ldots,V_k;\mathbb{R}) \end{equation*}

[1, Proposition 12.10].

References Link to heading

1. J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.