Twisted diagonal (category theory)

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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.