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Re #99: Thanks.
Re #100:
I’m not arguing for (i) over (ii), but I don’t see how (ii) in $[C^op,Set]$ is a way of talking about (i) in $C$. I don’t know what (i) in $C$ even means, since $C$ may not have a universe. There seems to be a serious misunderstanding about what (i) and (ii) are.
The partiality of (i) in $C$ is also external, so it doesn’t need a universe.
I reread some earlier comments on this subthread about (i) vs (ii). I think you’re right, and I misunderstood what Dan wants. It now seems to me like he might count natural families $\Pi X\,:\,Ob(C),\,(Hom(y X,Tm))_\bot$ as (i). And as far as I understand, this is essentially what you get from an element of $Tm_\bot$ constructed in the semantic logical framework without funny tricks.