# Very canonical constructions

You have two categories $$C_1$$ and $$C_2$$. We call a map of the classes $$\mathrm{Ob}(C_1)\rightarrow \mathrm{Ob}(C_2)$$ a construction. Sometimes you can find a functor $$C_1\rightarrow C_2$$ inducing this map, then you call your construction functorical or canonical.

Let us all a construction very canonical if there is a functor inducing it and between any two such functors there exists a natural isomorphism. What are some examples of very canonical constructions naturally arising in mathematics?

• Why the down-votes? Seems like a reasonable question to me (there even were some other questions on this site of similar flavour, e.g. mathoverflow.net/q/56938/140765, mathoverflow.net/q/19644/140765) – user140765 May 31 at 16:18
• I did not downvote or vote to close, but there is something a little strange about introducing a new definition seemingly "out of nowhere" and then asking others if it is non-vacuous. – Timothy Chow May 31 at 22:49
• @TimothyChow I agree, but I upvoted because this is a very canonical "out of nowhere" :). – Alec Rhea May 31 at 23:45
• Personally, I think this definition looks quite strange (and not at all "canonical", intuitively) from a categorical perspective: two functors inducing the same function on objects must in particular agree on objects; is the putative natural isomorphism between them required to also be the identity on objects? If so, you're just asking the two functors to be equal (i.e. unique); but if not, you're asking for a family of automorphisms of objects, which is strange. Also, uniqueness up to non-unique isomorphism is an unusual thing. – Mike Shulman Jun 2 at 16:00
• There are examples where these sorts of odd-looking things arise, like algebraic closures and more general Fraisse limits, or smothering functors in homotopy 2-categories, but it's not at all clear to me that this particular definition uses them in an interesting way. I would suggest to the OP that if they have an example in mind of this definition, they include it in the question; and otherwise, they start from the examples they are interested in and try to formulate a definition that captures them, rather than starting from a possibly-vacuous definition and looking for examples. – Mike Shulman Jun 2 at 16:04