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 < [Fe/H] < −0.9, in the Sculptor dwarf spheroidal galaxy. The chemical enrichment from AGB stars becomes apparent at [Fe/H] ≈ −2 in Sculptor, and causes [Y/Ba], [La/Ba], [Nd/Ba] and [Eu/Ba] to decrease with metallicity, reaching subsolar values at the highest [Fe/H] ≈ −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 [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 [Y/Ba]s/i = −0.85 ± 0.16, [La/Ba]s/i = −0.49 ± 0.17, and [Nd/Ba]s/i = −0.48 ± 0.12, 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 [Y/Ba]s/i and [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.