I think Double Categories [1] would be a more appropriate setting: In a double category, the vertical and horizontal arrows are of different types. In usual commutative diagrams, they are of the same type.

[1]: https://ncatlab.org/nlab/show/double+category

Commutative diagrams are to these squares what category theory is to analogy.

Why don't some phrases arrive at the same meaning? They don't commute.