TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A square commutes if and only if its transposed square commutes

Statement

Let $F ⊣ G$ be an adjunction between locally small categories. Suppose

$u : F c \to d, u^{'} : F c^{'} \to d^{'}, a : c \to c^{'}, b : d \to d^{'} .$

Then

$b \circ u = u^{'} \circ F (a) ⟺ G (b) \circ u^{♭} = (u^{'})^{♭} \circ a .$

The analogous equivalence holds after applying inverse transposition to a square between morphisms $c \to G d$ .

Facts & Assumptions

Given: The adjunction and the four typed morphisms in the Statement.

[L1]

Transposition is a bijection natural in both variables: for $a : c \to c^{'}$ , $b : d \to d^{'}$ and $v : F c^{'} \to d$ , one has $Φ_{c, d^{'}} (b \circ v \circ F (a)) = G (b) \circ Φ_{c^{'}, d} (v) \circ a$ (Under local smallness, transposition gives the natural hom-set bijection, and conversely).

Proof

technique · direct

1.1L1

If $b \circ u = u^{'} \circ F (a)$ , apply transposition. By [L1], the transpose of the left side is $G (b) \circ u^{♭}$ , while the transpose of the right side is $(u^{'})^{♭} \circ a$ , so the transposed square commutes.

1.2L1

Conversely, if the transposed square commutes, apply the inverse bijection to its two sides. The two inverse images are $b \circ u$ and $u^{'} \circ F (a)$ by [L1], so the original square commutes.

2.1step 1.1step 1.2L1∎

Repeating steps 1.1 and 1.2 with the inverse bijections proves the analogous assertion for inverse transposition.

Depends on

Under local smallness, transposition gives the natural hom-set bijection, and conversely

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Emily Riehl, Category Theory in Context, 2nd ed., Lemma 4.1.3 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Section 2.2 (standard reference, not scraped)