I try to learn the theory of category from The Joy of Cats. I got stacked with the first exercise (3A a).
If we have a simple graph with one vertex and 2 nodes all we know is that in category there is 2 objects and one homomorphism (beside identities). Hence:
\[C = (\{\{x, y\}, \{1, 2\}\}, \{f, id\}, id, \cdot), f = \{(x, 1), (y, 2)\}\]
and
\[C = (\{\{x\}, \{1\}, \{f, id\}, id, \cdot), f = \{(x, 1)\}\]
Should have the same graph but are they isomorphic?
Note about notation: I treat category as quadruple $(O, H, id, \cdot)$ where $O$ is class of objects (here is set), $H$ is class of homomorphisms (in example set). $id$ is identity and $\cdot$ is composition of homomorphisms.