The image of a functor need not be a subcategory Warning 1.2.19 gives an example when the image of a functor is not a subcategory:

But I'm confused: the author defines a functor $F$ right away without saying what the codomain category is. This causes the question: the image of that functor is not subcategory of which category? If the codomain is the category depicted on the right (which is the same as the image of $F$), then that is a subcategory of itself. 

The codomain is the category depicted on the right, and the image of $F$ is not a subcategory because it contains the morphisms $p$ and $q$ but not their composition $qp$. This is explained under the diagram.
