I suppose this depends a bit on what you are talking about (what kind of adaptation).
In human/social terms, they were coming into an area that already had lots of powerful tribes. They had to adapt to that situation. They did so by making themselves prized as mercenaries.
In environmental terms, they had to adapt to a new ecosystem, especially once they were driven out of Chapultepec to the shores of the lakes. Once there, their main adaptation was to make the chinampas -- the floating agricultural areas made by dredging mud from the lake bottoms.
Aztecs adapted extremely well to the Valley of Mexico. They borrowed culture, religion, technology of neighbors. They developed hydraulic agriculture; which led to a population explosion. They built causeways which connected islands with the mainland. They utilized a dam system and controlled lake levels; aqueducts brought fresh water. Food was grown in irrigated fields on artificial islands. They built impressive avenues, temples, and monuments. The Aztecs built a confederacy of neighboring city-states that began as tributary of Atzcapotzalco, later joined with Tecoco and Tlacopán to break away from Atzcapotzalco. Peasants worked plots, served in the army, merchants and craftsmen had guilds, exempt from military service, the Nobles (pipiltin) and priests controlled the land, but the priests were at the top of society. Farmers built dams, canals, terraced hillsides, drained swamps to increase the food supply while living in adobe brick houses.