TITLE: A category where maps are factorizations - what is this called?
QUESTION [13 upvotes]: Let $\mathcal C$ be a category, and define $\mathcal D$ to be the category whose objects are maps in $\mathcal C$, and where a map $f\to g$ is a factorization $pfq=g$.  Composition of $(p_1,q_1):f\to g$ and $(p_2,q_2):g\to h$ being $h=(p_2p_1)f(q_1q_2)$ and the obvious identity $f\to f$.
Does this category have a name?
For example: If $\mathcal C$ is a poset category, then $\mathcal D$ is just the closed intervals of the poset, ordered by inclusion.
$\mathcal D$ has the property that isomorphisms in the arrow category are isomorphisms in $\mathcal D$.

REPLY [13 votes]: This is called the twisted arrow category. I note with amusement that the linked definition uses exactly the same letters.