Are Unique up to Isomorphism?

To say that initial algebras are unique up to isomorphism means that they are all equivalent to each other with respect to the equivalence relation of isomorphism.

What does unique up to isomorphism mean?

Unique up to isomorphism means that all the objects satisfying a given definition are isomorphic, or have the same structure. Less formally, it means that they are the same object with different names for things.

What does unique up to mean?

This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, x is unique up to R means that all objects x under consideration are in the same equivalence class with respect to the relation R.