Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as $$g_{\mu\nu}+h_{\mu\nu}$$ or equivalently as $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$
The given metric has the property that $$g_{\mu\alpha}g^{\alpha\nu}=\delta^\nu_\mu$$and that $g_{\mu\nu}=g_{\nu\mu}$, which applies also to the metric after variation $g+h$.

Question: what is the relationship between $$\delta(g^{\mu\nu})$$ and $$h^{\mu\nu}$$
This is actually proving that for arbitrary tensors, $$A_{\mu\nu}B^{\mu\nu}=A^{\mu\nu}B_{\mu\nu}$$
How to prove this?