Foundation of mathematical objects modulo isomorphism in ZFC Suppose we want to define a notion of graph modulo isomorphism. The first thought that comes to mind is to consider the set of all graphs and then quotient it with respect to the equivalence relation induced by the notion of graph isomorphism, i.e. two graphs are equivalent if and only if they are isomorphic. A graph modulo isomorphism would be defined as an element of the quotient set. However, there is a small problem with this approach: the "set of all graphs" is actually not a set but a proper class in ZFC. And yet it seems pretty natural to me to consider two graphs as the same if they only differ by the name of their vertices and/or edges. We have the same problem if we restrict ourselves to finite graphs and also when we consider other mathematical structures such as groups or partial orders. Is there a way to overcome this issue and define standard mathematical objects modulo isomorphism in ZFC?

"Graphs modulo isomorphism" are called isomorphism types of graphs. Scott's trick is useful when the axiom of choice is unavailable. In ZFC (which includes the axiom of choice) every set is in bijection with some ordinal number, so you can simply define the isomorphism type of a graph $G$ to be the set of all graphs $H$, isomorphic to $G$, whose vertex set $V(H)$ is the least ordinal number in bijection with $V(G)$.
