We investigate the physical properties--such as the stellar mass, SFR, IR luminosity, X-ray luminosity, and hydrogen column density--of MIR galaxies and AGN at $z < 4$ in the 140 deg$^2$ field observed by SRG/eROSITA through the eFEDS survey. By cross-matching the WISE 22 $mu$m (W4)-detected sample and the eFEDS X-ray point-source catalog, we find that 692 extragalactic objects are detected by eROSITA. We have compiled a multiwavelength dataset. We have also performed (i) an X-ray spectral analysis, (ii) SED fitting using X-CIGALE, (iii) 2D image-decomposition analysis using Subaru HSC images, and (iv) optical spectral fitting with QSFit to investigate the AGN and host-galaxy properties. For 7,088 WISE W4 objects that are undetected by eROSITA, we have performed an X-ray stacking analysis to examine the typical physical properties of these X-ray faint and/or probably obscured objects. We find that (i) 82% of the eFEDS-W4 sources are classified as X-ray AGN with $log,L_{rm X} >$ 42 erg s$^{-1}$; (ii) 67% and 24% of the objects have $log,(L_{rm IR}/L_{odot}) > 12$ and 13, respectively; (iii) the relationship between $L_{rm X}$ and the 6 $mu$m luminosity is consistent with that reported in previous works; and (iv) the relationship between the Eddington ratio and $N_{rm H}$ for the eFEDS-W4 sample and a comparison with a model prediction from a galaxy-merger simulation indicates that approximately 5% of the eFEDS-W4 sources in our sample are likely to be in an AGN-feedback phase, in which strong radiation pressure from the AGN blows out the surrounding material from the nuclear region. Thanks to the wide area coverage of eFEDS, we have been able to constrain the ranges of the physical properties of the WISE W4 sample of AGNs at $z < 4$, providing a benchmark for forthcoming studies on a complete census of MIR galaxies selected from the full-depth eROSITA all-sky survey.
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