Does an initial algebra for a class have to belong to the class itself? In the context of algebraic data types, a concept of initial algebras is usually defined, e.g., in the following way:


  An algebra $S$ is initial in a class $C$ of algebras iff for every $A\in C$ there exists a unique homomorphlsm $h_A\colon S → A$.


At least "Initial Algebra Semantics and Continuous Algebras by" J. A. Goguen, J. W. Thatcher, E. G. Wagner, J. B. Wright states it this way. The proofs in that paper seem to imply that $S$ is assumed to belong to $C$ in the above definition.  Now, I'm wondering whether this restriction is meaningful. Put differently, has some reputable work defined an initial algerba for a class in such a way that the algebra is allowed to lie outside of the class?
Yes, the initial algebra is by definition one of the members of the class for which it is initial.

You may however be interested in the category-theoretic concept of a limit. Given a diagram (a collection of objects and morphisms between them), we can ask for an object which "best approximates" the diagram, but needs not be an element of the diagram. There are two such "best approximations", namely limit and colimit. The limit is like an initial algebra.