  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.
