More sophisticated distinction than "free" vs "derived"
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Presently we can treat the objects that are NOT the apex of a (co)cone as "real" / "immutable" data (cannot merge two elements, cannot add elements) whereas (co)limit objects are viewed as something more flexible - we might need more or fewer depending on how things shake out.
It's harder to make inferences about the derived objects. It seems like something is missing if we treat a product of two real objects A and B (with size |A|*|B|) as having a cardinality that can change.
We should keep track of which objects and morphisms are "determined". When all objects + morphisms in a diagram are 'determined', then the apex + legs become 'determined' as well.
addressed in the distinction between "frozen" and "unfrozen" objects