Like the previous Gaia releases, the Gaia EDR3 catalogue includes pre-computed cross-matches with optical/near infrared photometric and spectroscopic surveys. Combining the Gaia catalogue with other publicly available surveys obtained either from ground or from space closely meets the modern astronomical research request for combined surveys and/or wavelength domains. The main aim of adding a pre-computed cross-match to the Gaia is thus to complement Gaia with existing (and widely used by the scientific community) photometry and astrometry, allowing the full scientific exploitation of Gaia.
A cross-match algorithm is always a trade-off between multiple requirements (for example between completeness and correctness) and it should be chosen according to the scientific problem one is faced with. The catalogues matched with Gaia EDR3 are general surveys, not dedicated to a specific class of objects, and the cross-match algorithm should be general enough to be effectively used in different scientific cases.