Suppose that one had to consider (co)cones on a complicated diagram, with many arrows and objects and that one wished to prove that one of them is final/initial.
Given another (co)cone, one would construct a unique morphism such that the whole diagram commutes. Of course, one could always check commutativity for all "subdiagrams", but I assume this is far from necessary in most cases.
For example, for (co)cartesian squares, one only needs to show that two triangles commute.
In general, given a diagram, how would one determine the smallest number of equalities to check in order to show that it commutes?
In general, without making further assumptions on the category in question, there are no shortcuts: take any 2 objects then any two connecting paths must be equal.
Nevertheless...you can somewhat optimize this procedure by considering first pairs of objects joint by multiple - non intersecting - paths. Non intersecting paths are paths which have only the starting and end objects in common, and nothing in between. When you have k non intersecting paths joining 2 objects, then you need k-1 eqs. Once you have considered all such pairs of objects joint by multiple - non intersecting - paths, all other pairs are joint by intersecting paths - whose subpaths have been already considered and equated - plus maybe other non intersecting paths. You then simply add one more equation for each of these extra non intersecting path, and you are done. In short: start with the shortest parallel paths first.
I hope I have been clear enough, otherwise I will draw a picture.