# 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?