I have a category $C$ and will build a new category $X\_C$ out of it as follows. I hope this is a standard construction and that I can find somewhere in the literature its definition and properties worked out. Since I do not know the correct search string despite trying MacLane's book and the nlab, I ask here.


Each object of $X\_C$ is a sub-collection $(v\_{ij}:c\_i \to c\_j)$ of morphisms in $C$. The morphisms in $X\_C$ between $(v\_{ij}:c\_i \to c\_j)$ and $(w\_{k\ell}: d\_k \to d\_\ell)$ are collections of morphisms $x\_{ik} : c\_i \to d\_k$ of $C$ so that for any quadruple $(i,j,k,\ell)$ the obvious commutation relations hold, namely:


$$ x\_{j\ell}u\_{ij} = w\_{k\ell}x\_{ik} \text{ as morphisms }c\_i \to d\_\ell.$$


What is this $X\_C$ called? Thinking topologically, it looks like the "join" of the morphism diagrams, but google says nothing useful about "join categories" and the like. 


The motivation is as follows: my category $C$ consists of objects where each morphism represents a distance between source and target. Under certain compatibility conditions on morphisms (the obvious commutation relations), I can find a "witness" object within the desired distance of all objects in a collection. The next step is to see how these witnesses evolve as $C$ itself is transformed. For this I need a way to map compatible object collections to other compatible object collections.



Categorical nomenclature
It looks like the [free coproduct completion](http://ncatlab.org/toddtrimble/published/Small+cocompletions) of the [arrow category](http://ncatlab.org/nlab/show/arrow+category) of $C$ to me.


