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Looking for a proof different from the standard proof(s) of a result

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A better way to compute the mapping spaces of the category of spans in an enriched tensored ...

Not quite an answer, but I hope it helps: Your second definition looks (somewhat) like the usual definition of the V-valued hom of V-functors $[Sp,X] (a,b) = \int_{A \in Sp} X(aA, bA)$. If that's rig …
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