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I have finally split off dependent sum from dependent product. And added a few more paragraphs.
Added a section Relation to some limits with the remarks that dependent sum to the points preserves fiber products and that the naturality squares of the unit are pullbacks.
Is that a little confusing at dependent sum to have it be left adjoint to base change for any $f:A \to I$, then later consider only terminal maps?
Returning to my comment in #3, do type theorists use the terms dependent sum (or pair) and dependent product for adjoints to base change for general $f: A \to B$, or only for terminal maps?
Does anyone know the history of this? When did Martin-Löf first talk about dependent sum/product? Would he use the terms in the more general sense?
I donâ€™t know the history. In general, dependent sum/product are used in type theory for syntax that corresponds to adjoints to base change along a display map (i.e. to a map with terminal codomain, but in an arbitrary context). But since every map is isomorphic (in extensional type theory) or equivalent (in homotopy type theory) to a display map, in category theory we tend to use the words for adjoints to base change along an arbitrary map.
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