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.