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 $text{[Y/Ba]}_{s/i}=-0.85pm0.16$, $text{[La/Ba]}_{s/i}=-0.49pm0.17$, and $text{[Nd/Ba]}_{s/i}=-0.48pm0.12$, 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.
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