The eventual image

The paper’s Theorem 4.9 proves eventual image duality under a finite-type factorization system and identifies im_∞(f) simultaneously as a limit of the inverse image ladder and as a colimit of the direct image ladder. The candidate solution establishes the same theorem and identifications using a slightly different route: it first compares the limit/colimit of the constant iteration with the limit/colimit of the image ladders via axiom III/III*, then proves the limit-to-colimit comparison map is an isomorphism using equivariance under shift automorphisms and a retract argument. Aside from a minor imprecision about closure under retracts (standard for orthogonal factorization systems) and a small axiom-label slip in the paper (III vs III* in one line), both arguments are logically sound and mutually consistent.