gramatická analýza: Jsem Králík Jan, studující strojního inženýrství

Present active participle from studovat (3pl. studují plus -cí)

The main verb (the conjugated verb) is jsem; the subject of this sentence is therefore ; the predicate nominal (B in "A equals B" construction) is Králík Jan. Studující, refers to Králík Jan. (I am Králík Jan studying mechanical engineering, I am Králík Jan who studies mechanical engineering, I am Králík Jan, a student of mechanical engineering). The PrAP agrees with Králík Jan and thus is in the nominative case, masc. sg.

The direct object of studující is strojního inženýrství and it occurs in the genitive case (think of this as "a student of mechanical engineering)