Relative homotopy operations are well defined in their valid degrees

For n≥2, concatenate and reverse in coordinate 1. Its two faces are part of J; hence F2 makes the product and pasted representative homotopies continuous. The reparametrizations and reversal contractions in F3 act only on coordinate 1. Every J-face remains at x0, and the last-coordinate-zero face continues to map into A. Thus those same explicit homotopies prove associativity, unit and inverses in the relative set.

If n≥3, coordinates 1 and 2 are both available without changing the distinguished coordinate. The four-quarter identity and the two-unit calculation of F3 therefore apply to relative classes, proving commutativity. If n=2 only one coordinate is available, and if n=1 none is; the argument makes no stronger claim in those degrees.

Restriction to F sends a relative homotopy to a boundary-fixed homotopy in A, since $\mathrm{\partial }F\subseteq J$. For n≥2 it commutes pointwise with coordinate-1 concatenation, so $\mathrm{\partial }[a\ast b]=\mathrm{\partial }[a]\ast \mathrm{\partial }[b]$. For n=1 a relative homotopy moves the initial endpoint along a path in A, so its component is well-defined. Constant representatives map to the distinguished element in every degree.

For a map of based pairs $\varphi :(X,A,x_0)\to (Y,B,y_0)$, composing a representative or its homotopy with φ preserves all triple conditions. Composition and identity act pointwise, and composition commutes with products and with restriction to F. A based pair homotopy $\mathrm{\Phi }$ gives the representative homotopy $(u,t)\mapsto \mathrm{\Phi }(a(u),t)$; it sends F into B and J to y0. This proves all functoriality and homotopy assertions.