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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Loop suspension adjunction on based homotopy classes
Statement
Naturally for a well-pointed based CGWH space and a based CGWH space , there is a bijection , where is the kified compact-open space of loops based at .
Facts & Assumptions
Suspension is the quotient collapsing both ends and the basepoint track. Reduced cone suspension and cofiber sequence
Interval transposition preserves continuity, kification, based restrictions and homotopies. Interval exponential law and quotient homotopies
Proof
Given: The spaces, maps, and hypotheses in the statement above.
A based map pulls back to with A(x,0)=A(x,1)=y0 and A(x0,t)=y0. By F2 its transpose is continuous into the based loop space and maps x0 to the constant loop. Conversely a based b uncurries continuously and satisfies exactly those three equations, so descends to a based a by F1. Evaluation at (x,t) verifies both inverse identities.
Apply F2 with the additional homotopy parameter: the same formulas identify homotopies fixing the basepoint in either mapping set, and their inverses preserve both endpoint maps. Thus the map bijection descends to a bijection on based homotopy classes. Replacing x by f(x) or postcomposing each value with g commutes with evaluation, proving naturality in both variables.