Yes, the work was done while the others were on the ARIA project. Their goal was to provide a graded monad understanding of dl.acm.org/doi/10.1145/..., and related work. I'm not working on that now, but what we did is being written up and has nearly reached first draft phase.
Then set-indexed families of Psh(M), as Riley notes. But then this is Psh(M+), where the + adjoins an initial object.
Suddenly, we seemed very close to this section of Riley's thesis, but we didn't push things far in this direction:
My interest derived at first from the role of linear HoTT in the Sati-Schreiber program: ncatlab.org/schreiber/sh....
But then, oddly, working with Rajani and Orchard on a type theory for amortization, we found ourselves looking at models which were families of copresheaves on an ordered monoid.
I hope you find inspiration and an outlet. Nothing has come close for me to the thrill of writing on the n-Category Cafe c. 2006-2013.
Linked to "The unreasonable power of the lifting property in elementary mathematics"? Penultimate reference on 'nLab: factorization system':
ncatlab.org/nlab/show/fa...
The CW01 mentioned in the footnote is Caccamo & Winskel: A Higher-Order Calculus for Categories
www.brics.dk/RS/01/27/BRI.... It has rules such as
Maybe via ends/coends and this?:
arxiv.org/abs/1501.02503