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Full Idea
A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic.
Gist of Idea
Axioms are 'categorical' if all of their models are isomorphic
Source
Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2)
Book Ref
Colyvan,Mark: 'An Introduction to the Philosophy of Mathematics' [CUP 2012], p.25
A Reaction
The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'.