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?
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.