  Carbon abundances, especially at low metallicity, reveal the early chemical
evolution of a system, tracing the supernovae (SNe) that contributed and how
much of their ejecta made it into the next stellar generation. Our sample from
the \textit{Pristine} Inner Galaxy Survey (PIGS) includes $\sim 350$ metal-poor
([Fe/H]~$<-1.5$) stars in the main body of Sgr with good quality spectroscopic
observations. Our metal-poor Sgr population has a larger velocity dispersion
than metal-rich Sgr from the literature, which could be explained by outside-in
star formation, extreme Galactic tidal perturbations and/or the presence of a
metal-rich disc/bar $+$ a metal-poor halo. The average carbon abundance in Sgr
is similar to that of other classical dwarf galaxies (DGs) and consistently
lower than in the Milky Way by $\sim0.2-0.3$~dex at low metallicity. The
interstellar medium in DGs, including Sgr, may have retained yields from more
energetic Pop III and II SNe, thereby reducing the average [C/Fe].
Additionally, SNe~Ia, producing more Fe than C, would start to contribute at
lower metallicity in DGs/Sgr than in the Galaxy. The presence of a [C/Fe]
gradient for Sgr stars with [Fe/H]~$\gtrsim-2.0$ ($\sim 6.8\times 10^{-4}$ dex
arcmin$^{-1}$) suggests that SNe~Ia contributed in the system at those
metallicities, especially in its inner regions. There is a low frequency of
carbon-enhanced metal-poor (CEMP) stars in our Sgr sample. At higher
metallicity/carbon abundance (mostly CEMP-s) this may be due to photometric
selection effects, but those are less likely to affect CEMP-no stars. We
propose that, given the lower average [C/Fe] in DGs, using the same CEMP
definition ([C/Fe]~$>+0.7$) as in the Galaxy under-predicts the number of CEMP
stars in DGs, and for Sgr a cut at [C/Fe]$~\sim +0.35$ may be more appropriate,
which brings the frequency of CEMP stars in agreement with that in the Galaxy.
