1. Introduction
2. Motivation
3. The eROSITA Galaxy Cluster Survey and Its Simulated Twin
3.2. The eRASS1 Digital Twin
3.3. Matching Input and Detected Sources
3.4. Training and Test Samples
4. Selection Models with Surface Brightness Profiles
4.1. Illustration with a Single Feature: the $90^{\prime\prime}$ Cluster Counts
4.2. A Model Using Light Profiles from All Sources
4.3. Internal Validation of the Selection Models
5. Selection Models with Intermediate Variables
5.1. Mass and Redshift
5.2. Luminosity and Redshift
5.3. Unabsorbed Count Rate and Redshift
5.4. Internal Validation of the Selection Models
6. External Validation of the Selection Function
6.1. General Presentation of the Formalism
6.2. Testing the eRASS1 Cosmological Cluster Sample Against eFEDS
6.3. Testing the eFEDS Cluster Sample Against eRASS1 Cosmology Sample
6.4. Testing the eRASS1-Main Sample Against eFEDS
6.5. Testing the eRASS1 Cosmological Cluster Sample Against SPTpol-100d
6.6. Testing the SPTpol-100d Cluster Sample Against eRASS1-Cosmo
6.7. Testing the eRASS1 Cosmological Cluster Sample Against SPTpol-ECS
6.8. Testing the SPTpol-ECS Sample Against eRASS1 Cosmology Sample
7. Discussion
7.1. Linking Our Models with $\mathcal{L}_{\mathrm{ext}}$ and $\mathcal{L}_{\mathrm{det}}$
7.2. Underestimate of the eRASS1 Selection Model?
7.3. X-ray and SZ Correlated Detection
8. Conclusions