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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?

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    $\begingroup$ 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) $\endgroup$
    – user140765
    Commented May 31, 2019 at 16:18
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    $\begingroup$ 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. $\endgroup$
    – Timothy Chow
    Commented May 31, 2019 at 22:49
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    $\begingroup$ @TimothyChow I agree, but I upvoted because this is a very canonical "out of nowhere" :). $\endgroup$
    – Alec Rhea
    Commented May 31, 2019 at 23:45
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    $\begingroup$ 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. $\endgroup$
    – Mike Shulman
    Commented Jun 2, 2019 at 16:00
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    $\begingroup$ 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. $\endgroup$
    – Mike Shulman
    Commented Jun 2, 2019 at 16:04

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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.

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