In material set theory, we may speak of take and to be simply sets, rather than subsets of some ambient set? . Equivalently, one may take to be the class of all sets by default. (In this context, it’s important that whether and meet or are disjoint is independent of the ambient set or class.)
In constructive mathematics, the default meaning of ‘disjoint’ is as above, but sometimes one wants a definition relative to some inequality relation on . Then and are -disjoint if, whenever and , . (Ordinary disjointness is relative to the denial inequality.)
The concrete sets and are disjoint iff they have an internal disjoint union, in other words if their inclusions? into their union form a coproduct diagram in the category of sets. (Etymologically, of course, this is backwards.)
Many authors are unfamiliar with disjoint unions. When the disjoint union oid two abstract sets and is needed, they will typically lapse into material set theory (even when the work is otherwise perfectly structural), and make some comment such as ‘without loss of generality, assume that and are disjoint’ or (especially when ) ‘take two isomorphic copies of and ’, then call the disjoint union simply a ‘union’. (This works by the previous paragraph.)
To internalize the characterization in terms of internal disjoint unions is harder. If and have a join in the poset of subobjects , then we may ask whether this forms a coproduct diagram in . This should be equivalent if has disjoint coproducts.