Singular cohomology is contravariantly functorial

Define $f^{\mathrm{♯}}\phi =\phi f_{\mathrm{\#}}$ on cochains. It is additive and satisfies ${\delta }_X{f}^{\mathrm{♯}}\phi =\phi f_{\mathrm{\#}}{\mathrm{\partial }}_X=\phi {\mathrm{\partial }}_Y{f}_{\mathrm{\#}}=f^{\mathrm{♯}}{\delta }_Y\phi$ by [F1] and [F3]. Thus it takes cocycles to cocycles and a coboundary ${\delta }_Y\eta$ to ${\delta }_X(f^{\mathrm{♯}}\eta )$. It therefore induces the specified homomorphism on [F2] quotients, independently of every representative. In negative degrees use the unique map of zero groups.

On each singular simplex, $(g\circ f{)}_{\mathrm{\#}}\sigma =g\circ f\circ \sigma =g_{\mathrm{\#}}{f}_{\mathrm{\#}}\sigma$ and $(1_X{)}_{\mathrm{\#}}\sigma =\sigma$; extending linearly proves the same on chains. Consequently $\phi (g\circ f{)}_{\mathrm{\#}}=(\phi g_{\mathrm{\#}})f_{\mathrm{\#}}$ and $\phi (1_X{)}_{\mathrm{\#}}=\phi$ on cochains. These equalities also hold in negative degrees by the zero convention.

Postcomposition sends $\phi$ to $u\phi$. It is additive and $\delta (u\phi )=u\phi \mathrm{\partial }=u(\delta \phi )$. It preserves cocycles and coboundaries and induces $u_{\ast }$ on [F2]. Moreover $u(\phi f_{\mathrm{\#}})=(u\phi )f_{\mathrm{\#}}$, so the induced coefficient and space maps commute. Postcomposition by $1_G$ is identity, and postcomposition by $v\circ u$ equals successive postcomposition by $u$ and $v$, giving coefficient functoriality.

Step 1.1 supplies the quotient maps, and the cochain equalities of step 1.2 descend to their stated contravariant identity and composition laws. Step 1.3 proves coefficient functoriality and naturality. The formulas cover degree zero without quotient ambiguity since $B^0=0$, and negative degrees by zero maps. For empty source $X$, pullback lands in zero groups; a map to empty $Y$ exists only when $X$ is empty. Zero coefficients give zero groups. Identity maps on a point satisfy the same literal identity calculation. No representative or basis is selected and no AC is needed.