  The slow ($s$) and intermediate ($i$) neutron ($n$) capture processes occur
both in asymptotic giant branch (AGB) stars, and in massive stars. To study the
build-up of the $s$- and $i$-products at low metallicity, we investigate the
abundances of Y, Ba, La, Nd, and Eu in 98 stars, at $-2.4<\text{[Fe/H]}<-0.9$,
in the Sculptor dwarf spheroidal galaxy. The chemical enrichment from AGB stars
becomes apparent at $\text{[Fe/H]}\approx-2$ in Sculptor, and causes [Y/Ba],
[La/Ba], [Nd/Ba] and [Eu/Ba] to decrease with metallicity, reaching subsolar
values at the highest $\text{[Fe/H]}\approx-1$. To investigate individual
nucleosynthetic sites, we compared three $n$-rich Sculptor stars with
theoretical yields. One carbon-enhanced metal-poor (CEMP-no) star with high
$\text{[Sr, Y, Zr]}>+0.7$ is best fit with a model of a rapidly-rotating
massive star, the second (likely CH star) with the $i$-process, while the third
has no satisfactory fit. For a more general understanding of the build-up of
the heavy elements, we calculate for the first time the cumulative contribution
of the $s$- and $i$-processes to the chemical enrichment in Sculptor, and
compare with theoretical predictions. By correcting for the $r$-process, we
derive $\text{[Y/Ba]}_{s/i}=-0.85\pm0.16$, $\text{[La/Ba]}_{s/i}=-0.49\pm0.17$,
and $\text{[Nd/Ba]}_{s/i}=-0.48\pm0.12$, in the overall $s$- and/or $i$-process
in Sculptor. These abundance ratios are within the range of those of CEMP stars
in the Milky Way, which have either $s$- or $i$-process signatures. The low
$\text{[Y/Ba]}_{s/i}$ and $\text{[La/Ba]}_{s/i}$ that we measure in Sculptor
are inconsistent with them arising from the $s$-process only, but are more
compatible with models of the $i$-process. Thus we conclude that both the $s$-
and $i$-processes were important for the build-up of $n$-capture elements in
the Sculptor dwarf spheroidal galaxy.
