An employee's family members, regardless of their citizenship, have the right to accompany or join them in the country where the employee works. A family is defined as a husband/wife and children under the age of 21 (or older, if the employee provides for them), as well as the employee’s parents and the parents of the husband/wife, if the employee provides for them. As for the other family members, the country where the worker is employed is required to “consider favourably” their application so that they may join them.

Accompanying family members will receive a residence card or a document that has the same validity as that of the employee.

An employee’s husband/wife and children also have the right to work without restriction in the country where the employee is working, if they choose to do so.

Family members have the right to access general and vocational education available in the country of employment. An employee's children have the right to receive the same educational benefits available to other children in that country.

Obligations of EU citizens staying in Slovakia for more than 3 months

If an EU/EEA/Swiss citizen plans to stay in Slovakia for more than 3 months, they are also required to report the beginning of their stay to the relevant immigration police department within 10 business days of entering Slovakia, just as in the case of a short-term stay. They can then remain in Slovakia without any further obligations for up to 90 days from the date of entering the country. After this 90-day period has ended, an EU/EEA/Swiss citizen must apply for residence registration in Slovakia within 30 days.
https://www.researchinslovakia.saia.sk/en/main/welcome/visa-and-entry-conditions/prijimanie-cudzincov/vstup-a-pobyt/povinnosti-eu/ehp/svajciarsko/pobyt-na-slovensku-dlhsi-ako-3-mesiace