>>175 追加

IUTその4に下記説明ある
が、圏論で筋通した方が良さそう？

P72
Example 3.2. Categories. The notions of a [small] category and an isomorphism class of [covariant] functors between two given [small] categories yield an
example of a species. That is to say, at a set-theoretic level, one may think of a
[small] category as, for instance, a set of arrows, together with a set of composition
relations, that satisfies certain properties; one may think of a [covariant] functor
between [small] categories as the set given by the graph of the map on arrows determined by the functor [which satisfies certain properties]; one may think of an
isomorphism class of functors as a collection of such graphs, i.e., the graphs determined by the functors in the isomorphism class, which satisfies certain properties.
Then one has “dictionaries”
0-species ←→ the notion of a category
1-species ←→ the notion of an isomorphism class of functors
at the level of notions and
a 0-specimen ←→ a particular [small] category
a 1-specimen ←→ a particular isomorphism class of functors
at the level of specific mathematical objects in a specific ZFC-model. Moreover, one
verifies easily that species-isomorphisms between 0-species correspond to isomorphism classes of equivalences of categories in the usual sense.

Remark 3.2.1. Note that in the case of Example 3.2, one could also define a
notion of “2-species”, “2-specimens”, etc., via the notion of an “isomorphism of
functors”, and then take the 1-species under consideration to be the notion of a
functor [i.e., not an isomorphism class of functors]. Indeed, more generally, one
could define a notion of “n-species” for arbitrary integers n ? 1. Since, however,
this approach would only serve to add an unnecessary level of complexity to the
theory, we choose here to take the approach of working with “functors considered up to isomorphism”.