The coordinate connection of the flat chart is metric-selected

The operator $\mathrm{\nabla }$ is well defined on smooth fields because the component functions $Y^b$ of [L5] are smooth so $X(Y^b)$ is a smooth function. It is additive and $\mathbb{R}$-linear in $Y$ because the components do and $X$ is $\mathbb{R}$-linear; it is function-linear in $X$ because $(fX)(Y^b)=fX(Y^b)$; and ${\mathrm{\nabla }}_X(fY)={\sum }_{b}X(f{Y}^b){\mathrm{\partial }}_b={\sum }_b(X(f)Y^b+fX(Y^b)){\mathrm{\partial }}_b=X(f)Y+f{\mathrm{\nabla }}_{X}Y$. Hence $\mathrm{\nabla }$ is an affine connection by [L1].

For the coordinate fields, $Y={\mathrm{\partial }}_b$ has constant components $Y^c={\delta }^c{}_b$, so $X(Y^c)=0$ and the definition of step 1.1 gives ${\mathrm{\nabla }}_{{\mathrm{\partial }}_a}{\mathrm{\partial }}_b=0$. By [L2] the Christoffel symbols of $\mathrm{\nabla }$ in the global chart are ${\mathrm{\Gamma }}^c{}_{ab}=0$.

The metric components $A_{ab}=\mathrm{diag}(-1,1,1,1{)}_{ab}$ are constant, so ${\mathrm{\partial }}_a{A}_{bc}=0$; the compatibility identity of [L3] reads $0=0+0$ and holds, and the symmetry identity ${\mathrm{\Gamma }}^c{}_{ab}={\mathrm{\Gamma }}^c{}_{ba}$ reads $0=0$ and holds. By [L3] the connection is torsion-free and $A$-compatible, hence metric-selected by [L4].