Suppose, g is the metric tensor and F is a second rank mixed tensor. Then,
$$ \ g_{k \ i} * A_j^i = A_{k \ j} \ $$
Or,
$$\ g_{k \ i} * A_j^i = A_{j \ k} \ $$
In case of a mixed tensor, while contracting, we need to put the index ahead or behind. And what is the proof?
If $A$ is symmetric, it doesn't matter either way. If $A$ isn't symmetric, you have to indent one of the indices in $A^i_j$ to disambiguate. Explicitly$$A_{jk}=A_j^{\:i}g_{ik},\,A_{kj}=g_{ki}A^i_{\:j}.$$
Edit: let me make the indents bigger so you can see them:$$A_{jk}=A_j^{\quad i}g_{ik},\,A_{kj}=g_{ki}A^i_{\quad j}.$$Note that $A_j^{\quad i}\ne A^i_{\quad j}$.