Lemma: Naturality

Let $F$ be a diffeomorphism of $M$ equal to the identity outside a compact set. For all smooth Lorentzian $q,\bar{q}$ and smooth symmetric $u$,

as contravariant symmetric tensor densities.

Proof. $\sqrt{-q}R[q]$ and ${\mathfrak{L}}_{\mathrm{gf}}[u;\bar{q}]=\frac{1}{4}\sqrt{-\bar{q}}{\bar{q}}_{\mu \nu }{C^{\mu }}_{\bar{q}}[u]{C^{\nu }}_{\bar{q}}[u]$ are natural scalar densities: they are built from their arguments by tensor operations and the Levi-Civita connection of $\bar{q}$, so $\omega [F^{\ast }\text{fields}]=F^{\ast }\omega [\text{fields}]$. The argument of X-E's Proposition Covariance of the Euler operators applies verbatim: differentiate $\int \omega [F^{\ast }(\text{fields}+s\delta )]=\int \omega [\text{fields}+s\delta ]$ at $s=0$ for compactly supported variations $\delta$ of $q$, respectively of $u$, change variables by $F$ on the left, and use that the pairing of a tensor density with a tensor is invariant. The third identity follows from the first two by differentiating the first along $q=\bar{q}+su$ at $s=0$, since $F^{\ast }$ is linear.