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CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-05
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The underlying ordinary category is change of base along the underlying-hom functor

Statement

The underlying ordinary category construction is the special case of change of base along the lax monoidal functor V(1,−):V→Set.

Facts & Assumptions

Given: The change-of-base and underlying-category constructions.

[L1]

Lax monoidal change of base extends to a 2-functor on enriched categories (Change of base extends to enriched functors and natural transformations as a 2-functor).

[L2]

The underlying-category construction sends each hom-object to the set of global elements V(1,−) (The underlying-category construction is a 2-functor).

Proof

technique · direct
1.1

The hom-objects of the changed-base category along V(1,−) are exactly the sets V(1,A(A,B)), which are the hom-sets of A0 in [L2].

L1L2given
2.1

The composition and identity maps are also the same ones: the lax structure on V(1,−) is induced by tensoring global elements and then composing in V, which is exactly how [L2] defines composition and identities in the underlying category.

L2step 1.1
3.1

Therefore the underlying ordinary category is the change-of-base instance determined by V(1,−).

L1step 2.1∎

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources