Given a category $C$, is there a name for the following category:


* $\mathrm{Obj}(D) = \left\{ (x, y, f) \middle| x, y \in \mathrm{Obj}(C), f \in C(x, y) \right\}$
* $D((x, y, f), (x', y', f')) = \left\{ (g, h) \middle| g \in C(x', x), h \in C(y, y'), h \circ f \circ g = f' \right\}$


This is, if you want, the final category $D$ that has a covariant functor $P : D \to C$, a contravariant functor $Q : D \to C$ and a dinatural transformation $Q \to P$.


If this is a known construct, then I am also interested in higher-dimensional generalizations (i.e. if $C$ is an $n$-category, I would like an $n$-category $D$ with similar properties).



The source-side-opposite of the arrow category
As Zhen Lin suggests, what you are describing is called the "twisted arrow category" $C\_\#$ in Mac Lane's *Categories for the Working Mathematician* (see exercise IX.6.3, p.227). It goes by other names as well: for example "category of factorizations". A quick way of defining $C\_\#$ is as the category of elements of $hom\_C : C^{op} \times C \to \mathrm{Set}$ (in particular, the pairing of the projection functors is a discrete fibration $C\_\# \to C^{op} \times C$). You can find more discussion (including higher dimensional generalizations) [at the nlab](https://ncatlab.org/nlab/show/twisted+arrow+category).


