Categorical nomenclature

Question

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.

Answer 1

It looks like the free coproduct completion of the arrow category of $C$ to me.

Answer 2

I'm not sure if this is what you mean, but here are some terms that might come in handy to guide the search. If the collection $\{v_{ij}\}$ is always of the same shape, say with $i,j$ ranging over a fixed set, and is closed under composition, then what you're describing sounds like a functor category.

More precisely, such a category consists of diagrams $D$ of shape $J$. Here, $J$ is a category itself, with objects $i,j,\ldots$, and $D \colon J \to C$ is a functor. Explicitly, you can describe $D$ by giving all the objects $D(i)$ and all the morphisms $D(f)\colon D(i)\to D(j)$ for $f \colon i \to j$ in $J$. Your notation $c_i=D(i)$ suggests there is only one morphism $c_i \to c_j$. In that case $J$ has only one morphism $(i \leq j) \colon i \to j$, i.e. is a preorder $(J,\leq)$ regarded as a category; then $v_{ij} = D(i \leq j)$. Morphisms $D \to E$ are natural transformations. This category is usually denoted $C^J$.

The "witness" you mention at the end then sounds like a (co)limit of a diagram.