Similarly, a problem hard for a class C is called C-hard, e.g. NP-hard.
Normally, it is assumed that the reduction in question does not have higher computational complexity than the class itself.
Generally, complexity classes that have a recursive enumeration have known complete problems, whereas classes that lack a recursive enumeration have none.
For example, NP, co-NP, PLS, PPA all have known natural complete problems.
For example, Sipser showed that there is a language M such that BPPM (BPP with oracle M) has no complete problems.