One must make the distinction between constructing a model versus constructing the model. The latter is merely a hallucination. In more detail: when we state the axioms of, say, the naturals, we are merely listing some properties. One may have a strong feeling that these properties uniquely describe just one and only system, namley that of the natural numbers. In other words, one believes there is just one model for the axioms. But that is not true. Any model of the axioms will have to employ completely arbitrary names for the elements. Any formalism which produces a model must include an ad-hoc naming convention. Given a model you can simply change the names of the elements and obtain another model of the same axioms. Then, of course, one replaces the initial feeling of "there is a unique model" by "there is, up to isomorphism, a unique model". Now that is a totally different assertion.
So, if someone asks you if the naturals are a subset of the integers you should ask to first phrase the question correctly, since 'the naturals' and 'the integers' are illusions. That someone may then ask you: is there a model of the naturals in any model of the integeres, to which the answer is yes.