See universality class.
universal construction (having universal property), e.g. limits, representable object etc.
A terminal object is also called a universal terminal object, and an initial object is also called a universal initial object. There is also a notion of universal arrow.
While every colimit has a universal property, there is also an additional terminology. A colimit is a universal colimit if it is stable under pullbacks.
See also universe, Grothendieck universe, constructible universe.