A Riemannian metric has a unique Levi-Civita connection on the cotangent bundle

Fix a coordinate chart and let $(g_{ij})$ be the metric matrix there. Since $g$ is positive definite, $(g_{ij})$ is invertible at each point; write $(g^{ij})$ for the inverse matrix, define ${\mathrm{\Gamma }}_{ijk}:=\frac{1}{2}({\mathrm{\partial }}_i{g}_{jk}+{\mathrm{\partial }}_j{g}_{ik}-{\mathrm{\partial }}_k{g}_{ij})$ and ${\mathrm{\Gamma }}_{ij}^k:={\sum }_{\mathrm{\ell }}{g}^{k\mathrm{\ell }}{\mathrm{\Gamma }}_{ij\mathrm{\ell }}$, and use [F1] to obtain a local cotangent-bundle connection.

The formula for ${\mathrm{\Gamma }}_{ijk}$ is symmetric in $i$ and $j$, so ${\mathrm{\Gamma }}_{ij}^k={\mathrm{\Gamma }}_{ji}^k$. Hence the local connection is symmetric.

Adding ${\mathrm{\Gamma }}_{ijk}$ and ${\mathrm{\Gamma }}_{ikj}$ gives ${\mathrm{\Gamma }}_{ijk}+{\mathrm{\Gamma }}_{ikj}={\mathrm{\partial }}_i{g}_{jk}$, so the local connection is metric-compatible with $g$ in the sense of [F1].

Let $x=(x^i)$ and $y=(y^a)$ be overlapping charts, and let ${\mathrm{\nabla }}^x,{\mathrm{\nabla }}^y$ be the local connections from step 1.1 on those chart domains. On the overlap, write ${\mathrm{\partial }}_{y^b}={\sum }_i\frac{\mathrm{\partial }{x}^i}{\mathrm{\partial }{y}^b}{\mathrm{\partial }}_{x^i},d{y}^a={\sum }_k\frac{\mathrm{\partial }{y}^a}{\mathrm{\partial }{x}^k}d{x}^k\mathrm{.}$ Using the connection formula from [F1] and the Leibniz rule gives ${\mathrm{\nabla }}_{{\mathrm{\partial }}_{y^b}}^{x}d{y}^a=-{\sum }_c{\tilde{\mathrm{\Gamma }}}_{bc}^{a}d{y}^c,$ where ${\tilde{\mathrm{\Gamma }}}_{bc}^a={\sum }_{i,j,k}\frac{\mathrm{\partial }{y}^a}{\mathrm{\partial }{x}^k}{\mathrm{\Gamma }}_{ij}^k\frac{\mathrm{\partial }{x}^i}{\mathrm{\partial }{y}^b}\frac{\mathrm{\partial }{x}^j}{\mathrm{\partial }{y}^c}+{\sum }_k\frac{\mathrm{\partial }{y}^a}{\mathrm{\partial }{x}^k}\frac{{\mathrm{\partial }}^2{x}^k}{\mathrm{\partial }{y}^b\mathrm{\partial }{y}^c}\mathrm{.}$ Thus ${\mathrm{\nabla }}^x$ is also a cotangent-bundle connection in the $y$-chart, with coefficients ${\tilde{\mathrm{\Gamma }}}_{bc}^a$.

Conversely, let ${\tilde{\mathrm{\Gamma }}}_{ij}^k$ be the coefficients of any symmetric metric-compatible cotangent connection in this chart, and write ${\tilde{\mathrm{\Gamma }}}_{ijk}:={\sum }_{\mathrm{\ell }}{g}_{k\mathrm{\ell }}{\tilde{\mathrm{\Gamma }}}_{ij}^{\mathrm{\ell }}$. Symmetry and metric compatibility give ${\mathrm{\partial }}_i{g}_{jk}={\tilde{\mathrm{\Gamma }}}_{ijk}+{\tilde{\mathrm{\Gamma }}}_{ikj}$, ${\mathrm{\partial }}_j{g}_{ik}={\tilde{\mathrm{\Gamma }}}_{jik}+{\tilde{\mathrm{\Gamma }}}_{jki}$, and ${\mathrm{\partial }}_k{g}_{ij}={\tilde{\mathrm{\Gamma }}}_{kij}+{\tilde{\mathrm{\Gamma }}}_{kji}$; substituting ${\tilde{\mathrm{\Gamma }}}_{ijk}={\tilde{\mathrm{\Gamma }}}_{jik}$ and ${\tilde{\mathrm{\Gamma }}}_{ikj}={\tilde{\mathrm{\Gamma }}}_{kij}$ yields $2{\tilde{\mathrm{\Gamma }}}_{ijk}={\mathrm{\partial }}_i{g}_{jk}+{\mathrm{\partial }}_j{g}_{ik}-{\mathrm{\partial }}_k{g}_{ij}$, hence ${\tilde{\mathrm{\Gamma }}}_{ijk}={\mathrm{\Gamma }}_{ijk}$ and ${\tilde{\mathrm{\Gamma }}}_{ij}^k={\mathrm{\Gamma }}_{ij}^k$.

The coefficients from step 2.3 are symmetric in $b,c$: the first term is symmetric because ${\mathrm{\Gamma }}_{ij}^k={\mathrm{\Gamma }}_{ji}^k$ by step 2.1, and the second is symmetric by equality of mixed partials. If ${\widehat{g}}_{cd}={\sum }_{r,s}{g}_{rs}\frac{\mathrm{\partial }{x}^r}{\mathrm{\partial }{y}^c}\frac{\mathrm{\partial }{x}^s}{\mathrm{\partial }{y}^d}$ are the metric coefficients in the $y$-chart, then differentiating this identity and using step 2.2 in the $x$-chart yields $\frac{\mathrm{\partial }{\widehat{g}}_{cd}}{\mathrm{\partial }{y}^b}={\tilde{\mathrm{\Gamma }}}_{bcd}+{\tilde{\mathrm{\Gamma }}}_{bdc},$ where ${\tilde{\mathrm{\Gamma }}}_{bcd}:={\sum }_a{\widehat{g}}_{da}{\tilde{\mathrm{\Gamma }}}_{bc}^a$. So ${\mathrm{\nabla }}^x$ is symmetric and metric-compatible in the $y$-chart as well.

Step 3.1 now applies in the $y$-chart: the $y$-coefficients of ${\mathrm{\nabla }}^x$ are exactly the Christoffel symbols computed from the metric in that chart. But that is how ${\mathrm{\nabla }}^y$ was defined in step 1.1, so ${\mathrm{\nabla }}^x={\mathrm{\nabla }}^y$ on the overlap. Therefore the local operators glue to a global cotangent-bundle connection, and the same chartwise uniqueness proves global uniqueness.

Thus $g$ has a unique symmetric metric-compatible cotangent-bundle connection, whose coordinate coefficients are the displayed Christoffel symbols.