Let $C$ be a category, and let all objects $X_i, Y_j$ belong to $ob(C)$, and morphisms $f_{ij}, h_i, g_{ij}$ be morphisms between them in $C$.
Let us have a diagram then:
How do we verify it's commutative? I mean, the obvious way is to compare all compositions with domain $X_1$ and codomain $Y_4$, but there will be a lot of them.
Can we somehow reduce it to comparing compositions of certain subdiagrams? For example, we have a cube-shaped diagram here. Can we somehow check if the squares are commutative? Will that be enough, or no?
In general, how to deal with such diagrams(not necessarily resembling figures in $\mathbb{R}^3$, just any non-trivial, that is, harder than categorical product/coproduct universal property) diagrams?