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Let $X$ be an object in a category, and let $D$ be the empty diagram in the same category (containing no objects, and therefore no morphisms).

What should $\text{Hom}(D,X)$ be?

The only reasonable answers seem to me to be either one point or the empty set. A few sources I've looked at seem to imply that should be one point. Is this merely a convention, or is there a good natural reason to adopt this answer over the other?

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    $\begingroup$ What do you mean in general by a morphism between an object and a diagram, in either direction? $\endgroup$
    – Mike Shulman
    Commented Feb 28, 2021 at 21:40
  • $\begingroup$ The obvious definition seems to be a collection of morphisms, one $X \rightarrow Y$ for each object $Y$ of $D$, commuting with all of the arrows in the diagram. $\endgroup$
    – Kim
    Commented Mar 1, 2021 at 0:09
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    $\begingroup$ That's generally known as a cone under or over the diagram, and is equivalent to a natural transformation to/from a constant diagram as in Maxime's answer. Note that the definition directly yields the conclusion that there is exactly one cone over or under the empty diagram with any given vertex. $\endgroup$
    – Mike Shulman
    Commented Mar 1, 2021 at 0:32
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    $\begingroup$ Any time you ask for a collection of data, one datum for each $Y \in \emptyset$ [such that ...], you fill find that there is a unique such collection. $\endgroup$
    – Theo Johnson-Freyd
    Commented Mar 1, 2021 at 1:00
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    $\begingroup$ [As of this comment, there are currently two votes to close this question. This question should not be closed. It is a perfectly reasonable question that arises when one tries to understand the finer points of category theory.] $\endgroup$
    – Theo Johnson-Freyd
    Commented Mar 1, 2021 at 1:03

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$\emptyset$ is the initial category therefore $Fun(\emptyset, C)$ is the terminal category for any $C$.

So, if by "morphism to/from an object $X$ from/to the empty diagram $D$", you mean from/to the constant diagram at $X$, then both of these are indeed just a point. This is not a mere convention, but a calculation, in the same way that $\hom(\emptyset, X)$ only has one element for any set $X$.

If you mean something else, then you should make that more precise.

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    $\begingroup$ I would maybe not say "is reduced to a point...", but rather "has only one element...". It's not being 'reduced' from anything! :-) $\endgroup$
    – David Roberts ♦
    Commented Mar 1, 2021 at 1:09
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    $\begingroup$ @DavidRoberts : huh I thought it was standard terminology ! My bad, maybe it's only so in french :) $\endgroup$
    – Maxime Ramzi
    Commented Mar 1, 2021 at 8:20
  • $\begingroup$ Thanks for the edit (I'm thinking from a pedagogical angle, in particular, here) $\endgroup$
    – David Roberts ♦
    Commented Mar 1, 2021 at 10:49

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