Twisted Arrow Category

Let $\mathcal{C}$ be a category. A twisted arrow category of $\mathcal{C}$ is a catgory with:

• objects as morphisms in $\mathcal{C}$.
• morphisms between $f:a\to b$ and $g:c\to d$ as pairs of $\mathcal{C}$-morphisms $j:c\to a,k:b\to d$ such that the following diagram commutes:

In other words, a twisted arrow category is a variant of an arrow category in which the first morphism of the square in inverted.