How do you express that two commutative diagrams can be glued together along a common subgraph and retain commutativity?

Question

I'm wondering about this, because I want to write software that lets you operate on CD's on the computer screen.

I was wondering what existing mathematical tools are required to describe the titled operation?

For example, can we do it with the functors $F:I \to C, G:I \to C$ where $I$ is a finite indexing category, that is let $F, G$ be our two diagrams, how would we express that they share a common subdiagram in their images?

Answer 1

Commutative diagrams in $\mathcal C$ of shape $J$ are simply functors $J \to \mathcal C$. Therefore diagrams are objects in the category $\mathsf{Cat}/ \mathcal C$. In this category, coproducts corresponds to putting diagrams side by side; pullbacks corresponds to gluing two diagrams with a common part.

A more manageable version might be $\mathsf{FinCat}/\mathcal C$, that is diagrams with finitely many objects and morphisms.