[Loq]

> Is there a source for that narrative?

Girard mentions the connection with categories all the time.

For example in "Proofs and Types" he proves various theorems along the lines of: the sub-category of coherence spaces and stable maps (one of the main models that LL was developed for) induced by some LL fragment is cartesian closed category (IIRC). I think he developed LL in parts by fine-tuning it, until all the categories induced as models of fragments of LL have nice categorical properties. (To the extent that is possible.) When I was a PhD student, my supervisor suggested that I learn category theory to understand linear logic. (Not sure, in retrospect, that was the best course of action, but that was my trajectory)