Proof By Induction On Lists

I apologize–this is pretty vague and general but I am a curious sort and so I feel compelled to ask this.

From what little I know of category theory, it seems as if there should be some sort of category between the idea of Proof By Induction on Natural Number and Induction on Lists. I mean Prove Theorem T for 0 seems analogous to Prove Theorem T for empty list.

Is there anything to my naive intuition on this point?

That sounds like the idea of ornaments: lists are natural numbers ornamented with extra elements: