The first two say that and are ff and the second two say that they are disjoint subobjects of .
A coproduct is said to be universal if for any morphism , the pullbacks
exist and exhibit as a coproduct .
Finally, we say a 2-category is extensive if it has finite coproducts which are disjoint and universal. If it also has finite limits we say it is lextensive, and if it is also coherent we call it positive. (Note that disjoint coproducts in a coherent 2-category are always universal.)
An extensive 2-category does satisfy 2-categorical versions of the additional characterizations of an extensive 1-category, but unlike in the 1-categorical case, these alternate conditions do not seem to suffice to characterize extensivity. In particular, though, a 1-category is extensive as a 1-category iff it is so as a homwise-discrete 2-category.
If is extensive, so is , obviously. Less obvious is:
If is extensive, so are , , and . In other words, if is extensive, so is its -category of -truncated objects for .
Since the three given categories are closed in under limits and strict initial objects, it suffices to show they are closed under coproducts. First suppose given two morphisms . Then decomposes , and decomposes . Then the inclusions also decompose each . Now if there exists a 2-cell , it induces a map from each to the comma object of and . Since coproducts are disjoint and initials are strict, this implies that . Therefore, we have a decomposition so that and , where and .
Now, by universality of the coproduct , it follows that 2-cells are determined uniquely by pairs of 2-cells and . Therefore, if and are groupoidal, any 2-cells and are invertible, and thus so is any 2-cell ; so is groupoidal. And if and are posetal, any parallel 2-cells and are equal, and thus so are any ; so is posetal. And of course the discrete case follows by combining these.
However, the (0,1)-category (= poset) of (-1)-truncated objects (= subterminal objects) does not inherit extensivity, and in fact posets are almost never extensive: the only disjoint coproduct is .
We also have:
If is extensive, so are the fibrational slices and for any .
This is due to