安倍内閣は、第９０代の内閣総理大臣に指名された安倍晋三が２００６年９月２６日に発足させた内閣である。

、。

The Abe Cabinet was the cabinet launched on September 26, 2006, by Abe Shinzo, who was appointed as the 90th Prime Minister.

Literal

Abe Cabinet [topic-は], 90th-generation [possessive-の] Prime-Minister [to-に] nominated Abe Shinzo [subject-が] 2006 September 26th [on-に] launched cabinet [copula-である].

An encyclopedia-style definition sentence. 指名された ('was nominated/appointed') uses the passive for formal political language. 発足させた ('launched, established') is the causative of 発足する 'to start up' — the prime minister causes the cabinet to come into being. である closes as a formal copula.