A strict 2-equivalence between strict 2-categories is an equivalence of Cat-enriched categories. Thus, it consists of a pair of strict 2-functors and strict 2-natural isomorphisms? relating their composites to identities.
Traditionally this is called simply a 2-equivalence, with the word biequivalence used for the weaker notion. On the nLab we tend to say simply equivalence for the weaker notion, in line with our general philosophy that weak higher categories deserve the unadorned names.