## § An explanation for why permutations and linear orders are not naturally isomorphic

the number of linear orders on a finite set is the same as the number of bijections: the factorial of the cardinality. Every linear order on a set is isomorphic to any other, but a bijection is only isomorphic to another which has the same size and number of cycles. Thus, we have two functors $Perm: Set \rightarrow Set$ which sends a set to its set of permutations, and $Ord: Set \rightarrow Set$ which sends a set to its set of linear orders, such that the functors are equal on all objects (upto set isomorphism --- ie, produce outputs of the same size) but the functors fail to be isomorphic, since they have different criteria for "being equal".