In category theory in mathematics, the twisted diagonal of a category (also called the twisted arrow category), which makes the morphisms of a category into the objects of a new category, whose morphisms are then pairs of morphisms connecting domain and codomain with the twist coming from them being in opposite directions. It can be constructed as the category of elements of the Hom functor, which makes the twist come from the fact that it is contravariant in the first entry and covariant in the second entry. It can be generalized to the twisted diagonal of a simplicial set to which it corresponds under the nerve construction.
Definition.
For a category formula_1, its "twisted diagonal" formula_2 is a category, whose objects are its arrows:
 formula_3
and for which the morphisms between two such objects formula_4 and formula_5 are the pairs formula_6 and formula_7 of morphisms in formula_1 so that formula_9. If formula_10 denotes the category formula_11 with two objects and one non-trivial morphism (with the notation taken from the simplex category), then the twisted arrow category formula_2 is "not" the functor category formula_2 since the morphisms between the domains is reversed. An alternative definition is as the category of elements of the Hom functor:
 formula_14
There is a canonical functor:
 formula_15