Starting from finite-dimensional vector spaces (as objects) and the identity and dual functors, one can define a natural isomorphism, but this requires first adding additional structure, then restricting the maps from "all linear maps" to "linear maps that respect this structure". Explicitly, for each vector space, require that it comes with the data of an isomorphism to its dual, formula_190. In other words, take as objects vector spaces with a nondegenerate bilinear form formula_191. This defines an infranatural isomorphism (isomorphism for each object). One then restricts the maps to only those maps formula_192 that commute with the isomorphisms: formula_193 or in other words, preserve the bilinear form: formula_194. (These maps define the "naturalizer" of the isomorphisms.) The resulting category, with objects finite-dimensional vector spaces with a nondegenerate bilinear form, and maps linear transforms that respect the bilinear form, by construction has a natural isomorphism from the identity to the dual (each space has an isomorphism to its dual, and the maps in the category are required to commute). Viewed in this light, this construction (add transforms for each object, restrict maps to commute with these) is completely general, and does not depend on any particular properties of vector spaces.