# Is the image of a small category small?

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?

• 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. – Chessanator Mar 27 at 3:55
• I'm not sure what happens if $D$ isn't locally small, which is probably a more interesting question. – Chessanator Mar 27 at 3:56

## 1 Answer

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.

• 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. – Kevin Arlin Mar 27 at 5:13
• @KevinCarlson: I think you mean the underlying graph of the "naive" image of $F$ (which is not necessarily actually a subcategory). – Eric Wofsey Mar 27 at 5:16
• Sure, let's say that. – Kevin Arlin Mar 27 at 5:28