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The extended ROentgen Survey with an Imaging Telescope Array (eROSITA) onboard the Spectrum-Roentgen-Gamma (SRG) observatory is revolutionizing X-ray astronomy. It provides large samples of active galactic nuclei (AGN) and clusters of galaxies, with the potential of studying X-ray sources and measuring cosmological parameters using X-ray-selected samples with higher precision than ever before. We aim to study the detection, and the selection of AGN and clusters in the first eROSITA all-sky survey (eRASS1), and to characterize the properties of the source catalog. We produced a half-sky eRASS1 simulation, by combining models that truthfully represent the clusters and AGN. In total, we simulated 1 116 758 clusters and 225 583 320 AGN. We ran the standard eROSITA detection algorithm. We matched the input and source catalogs with a photon-based algorithm. We perfectly recovered the bright sources. We detected half of the AGN brighter than 2e-14 erg/s/cm2 as point sources and half of the clusters brighter than 3e-13 erg/s/cm2 as extended in the 0.5-2.0 keV band. We quantified the detection performance in terms of completeness, false detection rate, and contamination. We studied the source catalog according to multiple cuts of detection and extension likelihood. We find that the detection of clusters is mainly driven by flux and exposure. It depends on secondary effects, such as the size of the clusters and their dynamical state. The cool core bias mostly affects faint clusters classified as point sources, while its impact on the extent-selected sample is small. The measured X-ray luminosity of the detected clusters is compatible with the simulated values. We discuss how to best build samples of galaxy clusters for cosmological purposes, accounting for the nonuniform depth of eROSITA. This simulation provides a digital twin of the real eRASS1.