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To begin with, we consider an arbitrary functional $F[\gamma_{\mu\nu}, {\cal A}^a_\mu, {\cal C}^\mu, \bar{{\cal C}}_\mu, \Sigma^a, \bar{\Sigma}^a]$ of the dynamical fields at hand. At this stage the splitting into fluctuations and background configurations has not yet been performed. An infinitesimal gauge transformation of $F$, considered a scalar functional of its arguments, consists of a diffeomorphism along $v^\mu$ and an $SU(N)$ transformation with parameters $\lambda^a$. It can be implemented as $$\begin{aligned}
&F[\gamma + \delta_{\rm D} (v)\gamma + \delta_{\rm YM} (\lambda)\gamma , {\cal A} + \delta_{\rm D} (v){\cal A} + \delta_{\rm YM} (\lambda){\cal A}, {\cal C} + \delta_{\rm D} (v){\cal C} + \delta_{\rm YM} (\lambda){\cal C},\\
&\hspace{0.7cm}\bar{{\cal C}} + \delta_{\rm D} (v)\bar{{\cal C}} + \delta_{\rm YM} (\lambda)\bar{{\cal C}}, \Sigma + \delta_{\rm D} (v){\Sigma} + \delta_{\rm YM} (\lambda){\Sigma}, \bar{\Sigma} + \delta_{\rm D} (v)\bar{\Sigma} + \delta_{\rm YM} (\lambda)\bar{\Sigma}] \\
&= F[\gamma, {\cal A}, {\cal C}, \bar{{\cal C}}, \Sigma, \bar{\Sigma}] - {\cal W}_{\rm D}(v)F[\gamma, {\cal A}, {\cal C}, \bar{{\cal C}}, \Sigma, \bar{\Sigma}] - {\cal W}_{\rm YM}(\lambda)F[\gamma, {\cal A}, {\cal C}, \bar{{\cal C}}, \Sigma, \bar{\Sigma}]\\
&\hspace{0.5cm} + {\cal O} (v^2, v \lambda, \lambda^2)
\end{aligned}$$

with the corresponding Ward operators generating diffeomorphisms, $$\begin{aligned}
{\cal W}_{\rm D}(v) &\equiv -\int\:{\rm d}^d x\:\bigg(\delta_{\rm D}(v)\gamma_{\mu\nu} (x)\frac{\delta}{\delta \gamma_{\mu\nu}(x)} + \delta_{\rm D}(v){\cal 