By considering the various CEMP subclasses separately, we try to derive, from the specific signatures imprinted on the abundances, parameters (such as metallicity, mass, temperature, and neutron source) characterizing AGB nucleosynthesis from the specific signatures imprinted on the abundances, and separate them from the impact of thermohaline mixing, first dredge-up, and dilution associated with the mass transfer from the companion.To put CEMP stars in a broad context, we collect abundances for about 180 stars of various metallicities, luminosity classes, and abundance patterns, from our own sample and from literature. First, we show that there are CEMP stars which share the properties of CEMP-s stars and CEMP-no stars (which we call CEMP-low-s stars). We also show that there is a strong correlation between Ba and C abundances in the s-only CEMP stars. This strongly points at the operation of the 13C neutron source in low-mass AGB stars. For the CEMP-rs stars (seemingly enriched with elements from both the s- and r-processes), the correlation of the N abundances with abundances of heavy elements from the 2nd and 3rd s-process peaks bears instead the signature of the 22Ne neutron source. Adding the fact that CEMP-rs stars exhibit O and Mg enhancements, we conclude that extremely hot conditions prevailed during the thermal pulses of the contaminating AGB stars. Finally, we argue that most CEMP-no stars (with no overabundances for the neutron-capture elements) are likely the extremely metal-poor counterparts of CEMP neutron-capture-rich stars. We also show that the C enhancement in CEMP-no stars declines with metallicity at extremely low metallicity ([Fe/H]~< -3.2). This trend is not predicted by any of the current AGB models.
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