I was wondering if there is some way to codify the mapping of the bottom-up description of a concrete category to its universal constructions.
You should definitely look into the code of Saul Youssef. It's quite involved and doesn't currently compile. But maybe you can extract some interesting ideas out of it.
It lacks documentation, but if you know a bit of category, some the constructions seem pretty natural.
I don't however know whether Youssef's code is helpful for what you have in mind.
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