Relating commutativity of binary operators and diagrams

Does that imply relating binary operators and diagrams more generally?

If a diagram commutes we can say g.f = h.

If I am chasing a diagram in a category and arrive at a limit of a sub-diagram, I have to choose a path. This can be done using a semigroup. If the pathways are commutative, then I can use a trivial semigroup, and the operation commutes.

a = b -> fst (a, b) = fst (b, a)

In some sense, saying that a diagram commutes is a weaker statement than saying two paths are equal, but strictly in the context of choosing a path using a free semigroup, commutativity and equality are one and the same.

Is equality an initial object in some category of binary relations? Is it a free construction in a category of commutative binary relations?