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Let $C$ be a small category, let $D$ be a locally small category. Given a functor $F:C\to D$, the image of $F$ may not be a category. Now following the nLab, let's instead call the image of $F$ the subcategory of $D$ generated by the images of the objects and morphisms of $C$ under $F$, i.e. close it under composition.

With this notion of image, by construction we have a subcategory of $D$. Is this subcategory again small?

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  • $\begingroup$ Since $D$ is locally small, the image of $F$ should have a set of objects (the image of the set of objects of $C$, since $C$ is small and you can't reach any object outside that image) and each hom is a set (as a subset of the hom-set of locally-small $D$) so the image is small. $\endgroup$ – Chessanator Mar 27 at 3:55
  • $\begingroup$ I'm not sure what happens if $D$ isn't locally small, which is probably a more interesting question. $\endgroup$ – Chessanator Mar 27 at 3:56
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Yes, this is trivial. Every morphism in the image of $F$ can be written as a finite composition of morphisms that come from morphisms of $C$, and there is only a small set of such finite compositions.

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    $\begingroup$ Another way to put this is that the image of $F$ is a quotient of the category freely generated by the underlying graph of $C$, which certainly remains small. $\endgroup$ – Kevin Arlin Mar 27 at 5:13
  • $\begingroup$ @KevinCarlson: I think you mean the underlying graph of the "naive" image of $F$ (which is not necessarily actually a subcategory). $\endgroup$ – Eric Wofsey Mar 27 at 5:16
  • $\begingroup$ Sure, let's say that. $\endgroup$ – Kevin Arlin Mar 27 at 5:28

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