The twisted arrow category and its projection to ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$

Let $\mathcal{C}$ be a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). Its twisted arrow category $\mathrm{Tw}(\mathcal{C})$ has the following data. Its objects are the morphisms of $\mathcal{C}$. For objects $f:c\to c^{\mathrm{\prime }}$ and $g:d\to d^{\mathrm{\prime }}$, a morphism $f\to g$ is a pair $(a,b)$ with $bfa=g$, where $a:d\to c$ and $b:c^{\mathrm{\prime }}\to d^{\mathrm{\prime }}$ are morphisms of $\mathcal{C}$; the identity of $f$ is $(1_c,1_{c^{\mathrm{\prime }}})$, and the composite of $(a,b):f\to g$ with $(a^{\mathrm{\prime }},b^{\mathrm{\prime }}):g\to h$ is

$(a^{\mathrm{\prime }},b^{\mathrm{\prime }})\circ (a,b):=(a\circ a^{\mathrm{\prime }},b^{\mathrm{\prime }}\circ b):f\to h\mathrm{.}$

That composite is a morphism $f\to h$ because $(b^{\mathrm{\prime }}b)f(a{a}^{\mathrm{\prime }})=b^{\mathrm{\prime }}(bfa)a^{\mathrm{\prime }}=b^{\mathrm{\prime }}g{a}^{\mathrm{\prime }}=h$, associativity and the identity laws are inherited from $\mathcal{C}$, and the composite of the two identities is the identity, so these data satisfy Category, object, morphism, domain, codomain, identity, composition, and hom-collection.

The twisted arrow projection is the assignment

$\pi :\mathrm{Tw}(\mathcal{C})⟶{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C},(f:c\to c^{\mathrm{\prime }})⟼(c,c^{\mathrm{\prime }}),(a,b)⟼(a,b),$

where in the target the first coordinate $a$ is read as the morphism $c\to d$ of ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ corresponding to $a:d\to c$ (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$, Product category and its projection functors). It preserves identities by construction, and it preserves composites because composition in ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ is componentwise with the first coordinate reversed, which is the order written in the display above; so $\pi$ is a functor (Covariant functor, identity functor, composite functor, and contravariant functor).

The name records the twist: a morphism of $\mathrm{Tw}(\mathcal{C})$ acts by precomposition in one coordinate and by postcomposition in the other, so the first coordinate runs backwards. That is exactly what makes $\pi$ land in ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ rather than in $\mathcal{C}\times \mathcal{C}$, and it is why a diagram indexed by $\mathrm{Tw}(\mathcal{C})$ can see a functor of two variables of opposite variance.

The opposite orientation is also in use in the literature, with $\mathrm{Tw}(\mathcal{C})$ naming what is here $\mathrm{Tw}(\mathcal{C}{)}^{\mathrm{o}\mathrm{p}}$. The orientation fixed above is in force everywhere on this page, and Orientation and notation conventions in force on this page states it alongside the integral conventions; every statement below that names $\mathrm{Tw}(\mathcal{C})$ is to be read with the definition given here and is false under the other one.