Is there a term for the inverse of a fixed-point operator?

When working with recursion it is often useful to find the least or greatest fixed points of a morphism, often using a fixed-point combinator. When working with recursion schemes, the inverse operation — finding a morphism for which a value is a fixed point — is also very useful, in which case the operation is sometimes called parameterizing and its result sometimes called a pattern functor or base functor.

Is there a more general term for this inverse fixed-point operation or its resulting morphism?