I've started reading Abstract and Concrete Categories: The Joy of Cats to learn about Category Theory, and I've got stuck on one of the first exercises. I have a background in Computer Science, and to be honest I'm not too familiar with many of the concrete categories taken from other areas of Mathematics that the book uses as examples, and I'm also not aware about any relevant results/theorems from other topics that might help me here.
I've read the first chapter of the book, and I decided to do some of the first exercises to test my understanding of the material. One of the exercises that has been giving me trouble is the following: Exercises 3A(b). Basically I need to find two non-isomorphic categories with the same graph (where the definition of graph given in the exercise looks more like that of a multigraph/quiver). Now, the way I understood that definition of the graph $G(A) = (V,E,d,c)$ of a category $A = (O,hom,id,\circ)$ there seems to be a 1-1 correspondence between vertices in $V$ and objects in a $O$, and between edges in $E$ and morphisms in $hom$, so it would seem to be impossible to have two different categories with the same graph (or even with isomorphic graphs for that matter). Now, I'm sure I am just reading it wrong and that the exercise probably has an easy answer. Can you help me figure out what I'm missing here?
I've done some research prior to posting this question, and the most similar question I could find were Need help discerning category isomorphism, Is a graph determined up to isomorphism by its path category?, Show that for each of the following graphs G there exists up to isomorphism precisely one category A with G(A) = G., and Thin categories and graphs (isomorphism of categories). Unfortunately the first one doesn't really clear my doubts, since I would argue that the categories laid out in the first answer don't have the same quiver. The only answer to the second one wasn't very satisfactory for me either, and the last two address other subparts of exercise 3A, but none of them helped me in understanding what I am doing wrong for part $(b)$.