Let $\Delta$ be the simplex category. For $m \leq c \leq n$, let $[m] \to [c] \to [n]$ be the composition of two injective morphisms in $\Delta$.

We now define a category $\Box$ with same objects as $\Delta$, but we think of them as cubes instead of simplices. Combinatorially, the $m$-facets of the $n$-cube $[n]$ should be exactly compositions of the form $[m] \to [c] \to [n]$ as above. Thus, I would like to define

$$\Box_m([m],[n]) := \coprod_{c=0}^\infty ( \Delta_m([c],[n]) \times \Delta_m([m],[c]) ), $$ where the index $m$ indicates that I am considering only monomorphisms. I'm not a category theory expert, but this looks a lot like a coend over the product of hom-functors: $$F(\cdot,\cdot):=( \Delta_m(\cdot,[n]) \times \Delta_m([m],\cdot) ) \qquad \Rightarrow \qquad \Box_m([m],[n]) := \int^{[c] \in \Delta} F([c],[c]).$$ My question is basically, if this observation - i.e. redefining the morphism set in a category by taking a coend of the previous form - is helpful in the following sense:

1. It would be nice if the above definition would automatically give me a composition law by some categorical nonsense
2. It would also be nice if the above definition would guide the way for defining general morphisms (simply omit '$m$').

Both points can of course be done directly without a lot of effort. Still it would be a nice confirmation if the construction turns out to be 'natural' in the categorical sense.