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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?

Загнивающий Запад
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    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. https://mathoverflow.net/q/56938/140765, https://mathoverflow.net/q/19644/140765) –  May 31 '19 at 16:18
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    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 '19 at 22:49
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    @TimothyChow I agree, but I upvoted because this is a very canonical "out of nowhere" :). – Alec Rhea May 31 '19 at 23:45
  • Generally the onus is on whoever introduces a definition to give at least one example showing that the definition is interesting. The other questions about words like 'canonical' and 'natural' were trying to understand how other mathematicians use the word. – Mike Shulman Jun 02 '19 at 15:59
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    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 02 '19 at 16:00
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    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 02 '19 at 16:04
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    I asked a similar question in https://mathoverflow.net/q/332597/24563 for a different notion and nobody downvoted... Of course I have examples and I did not include them in the post because I did not want to influence the intuition of readers. Maybe the OP has the same motivation ? – Philippe Gaucher Jun 04 '19 at 07:59
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    The identity on objects for the category of sets should be an example. Not sure it is very helpful though. – Philippe Gaucher Jun 04 '19 at 08:04

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Such constructions on the category of sets were studied by Barkhudaryan, El Bashir, and Trnkova. They call them "DVO functors" (for "Defined by Values on Objects").

It appears there are a few papers citing this paper which study similar questions, as can be found by a Google Scholar search.

Tim Campion
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