Context : This paper is part of a larger project in which we want to focus on the still poorly understood AGB third dredge-up processes and associated s-process nucleosynthesis .

Aims : We confront accurate spectral abundance analyses of post-AGB stars in both the LMC and SMC , to state-of-the-art AGB model predictions .
With this comparison we aim at improving our understanding of the 3rd dredge-up phenomena and their dependencies on initial mass and metallicity .

Methods : Because of the well constrained distance with respect to Galactic post-AGB stars , we choose an extra-galactic post-AGB star for this contribution , namely the only known 21 \mu m object of the Small Magellanic Cloud ( SMC ) : J004441.04-732136.4 .
We used optical UVES spectra to perform an accurate spectral abundance analysis .
With photometric data of multiple catalogues we construct a spectral energy distribution and perform a variability analysis .
The results are then compared to predictions of tailored theoretical chemical AGB evolutionary models for which we used two evolution codes .

Results : Spectral abundance results reveal J004441.04-732136.4 to be one of the most s-process enriched objects found up to date , while the photospheric C/O ratio of 1.9 \pm 0.7 , shows the star is only modestly C-rich .
J004441.04-732136.4 also displays a low [ \textrm { Fe / H } ] = -1.34 \pm 0.32 , which is significantly lower than the mean metallicity of the SMC .
From the SED , a luminosity of 7600 \pm 200 \textrm { L } _ { \odot } is found , together with E ( B-V ) = 0.64 \pm 0.02 .
According to evolutionary post-AGB tracks , the initial mass should be \approx 1.3 \textrm { M } _ { \odot } .
The photometric variability shows a clear period of 97.6 \pm 0.3 days .
The detected C/O as well as the high s-process overabundances ( e.g .
[ Y/Fe ] = 2.15 , [ La/Fe ] = 2.84 ) are hard to reconcile with the predictions .
The chemical models also predict a high Pb abundance , which is not compatible with the detected spectrum , and a very high ^ { 12 } C/ ^ { 13 } C , which is not yet constrained by observations .
The predictions are only marginally dependent on the evolution codes used .

Conclusions : By virtue of their spectral types , favourable bolometric corrections as well as their constrained distances , post-AGB stars in external galaxies offer unprecedented tests to AGB nucleosynthesis and dredge-up predictions .
We focus here on one object J004441.04-732136.4 , which is the only known 21 \mu m source of the SMC .
We show that our theoretical predictions match the s-process distribution , but fail in reproducing the detected high overabundances and predict a high Pb abundance which is not detected .
Additionally , there remain serious problems in explaining the observed pulsational properties of this source .