OP. This is what Grok says about the theorem:

This excerpt from algebraic topology (likely John Hatcher’s Algebraic Topology, Ch. 3) sets up tools for deriving the long exact cohomology sequence of a pair of spaces ((X, A)).
In brief:
• It defines inclusion maps (i) and a weak retraction (r) between “cylinder-like” product spaces (e.g., (A \times I \times X \times {i}) to (A \times I \times X \times {1}), where (I = [0,1]) is the unit interval).
• Using exactness of the cohomology sequence for these spaces, any cohomology class (u \in H^q(A \times I \times X \times {i})) lifts uniquely to (u’ \in H^q(A \times I \times X \times {i}, A \times I \times X \times {1})) satisfying (u = i^* u’ + \partial i^* u) (boundary terms).
• A map (h: (X, A) \to (A \times I \times X \times {i}, A \times I \times X \times {1})) induces an iso (h^), yielding a natural epimorphism (transformation) (\Delta: H^q(A \times I \times X \times {i}) \to H^q(X, A)) via (\Delta(u) = h^ u’).
• Lemma 4 proves commutativity (up to sign ((-1)^{q+1})) of a triangle diagram linking (\Delta), a connecting homomorphism (\delta: H^q(A \times I \times X \times {i}) \to H^{q-1}(X, A) \times (I, I)), and another boundary map (\tau: H^{q-1}(X, A) \times (I, I) \to H^q(X, A))—essential for exactness in pair cohomology sequences. The proof mirrors an earlier homotopy argument (Thm. 8.2.4).

I guess I’m not white.