However, unlike a set, an object of a general category need not be determined by its global elements (not even up to isomorphism).
In an elementary topos the global elements of the subobject classifier Ω form a Heyting algebra when ordered by inclusion of the corresponding subobjects of the terminal object.
[3] For example, Grph happens to be a topos, whose subobject classifier Ω is a two-vertex directed clique with an additional self-loop (so five edges, three of which are self-loops and hence the global elements of Ω).
The internal logic of Grph is therefore based on the three-element Heyting algebra as its truth values.
That is, for each pair of distinct arrows A → B in the category, there should exist a global element whose compositions with them are different from each other.