Context: The physical origin behind organic emission in embedded low-mass star formation has been fiercely debated in the last two decades. A multitude of scenarios have been proposed, from a hot corino to PDRs on cavity walls to shock excitation. Aims: The aim of this paper is to determine the location and the corresponding physical conditions of the gas responsible for organics emission lines. The outflows around the small protocluster NGC 2071 are an ideal testbed to differentiate between various scenarios. Methods: Using Herschel-HIFI and the SMA, observations of CH3OH, H2CO and CH3CN emission lines over a wide range of excitation energies were obtained. Comparisons to a grid of radiative transfer models provide constraints on the physical conditions. Comparison to H2O line shape is able to trace gas-phase synthesis versus a sputtered origin. Results: Emission of organics originates in three spots: the continuum sources IRS 1 (B) and IRS 3 (A) as well as a outflow position (F). Densities are above 10$^7$ cm$^{-3}$ and temperatures between 100 to 200 K. CH3OH emission observed with HIFI originates in all three regions and cannot be associated with a single region. Very little organic emission originates outside of these regions. Conclusions: Although the three regions are small (<1,500 AU), gas-phase organics likely originate from sputtering of ices due to outflow activity. The derived high densities (>10$^7$ cm$^{-3}$) are likely a requirement for organic molecules to survive from being destroyed by shock products. The lack of spatially extended emission confirms that organic molecules cannot (re)form through gas-phase synthesis, as opposed to H2O, which shows strong line wing emission. The lack of CH3CN emission at F is evidence for a different history of ice processing due to the absence of a protostar at that location and recent ice mantle evaporation.
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