Consider a (somehow naturally) specified mapping from one class of structures into another (or the same) class of structures, e.g. the mapping of a
- graph/group to its automorphism group,
- group to its Cayley graph,
- graph to its complement graph,
- group to its opposite group,
- category to its opposite category
- graph to its line graph,
- planar graph to its dual graph,
- and so on
Is there a somehow standarized way to detect whether such a mapping can be made a functor?
A prerequisite is to equip both sides (classes) of the mapping with morphisms to make them categories. If one has achieved this - usually in one of many possible ways - one has to look for specific mappings of morphisms in the source category to morphisms in the target category that fulfill the functor conditions.
Given such morphisms and a mapping of morphisms one may be able to check whether they fulfill the functor conditions, but - given only the mapping between objects -
how to find appropriate morphisms and their mapping, resp. how to show that you cannot find them?
Only some of the examples above give rise to functors, but I am especially insecure about the last two examples, so I'd like to focus the question:
How do you show that a mapping between structures can not be made a functor?