sort; see [@Vokrinek] for details.

Thus, although non-equivariant effective homology is not the same as equivariant effective homology, it is possible to construct one from the other. In this paper, effective homology will be always understood in the equivariant sense.

We recall that the *Eilenberg–Zilber reduction* is a particular reduction $C_*(X\times Y){\xRightarrow{\ \ }}C_*X\otimes C_*Y$; see e.g. [@EML54; @polypost; @SergerGenova]. It is known to be functorial (see e.g. [@EML54 Theorem 2.1a]), and hence it is equivariant. We extend it to pairs.

\[prop:relative\_product\] If pairs $(X,A)$ and $(Y,B)$ of simplicial sets are equipped with effective homology, then it is also possible to equip the pair $$(X,A)\times(Y,B){\stackrel{\mathrm{def}}{=}}\big(X\times Y,(A\times Y)\cup(X\times B)\big)$$ with effective homology.

The Eilenberg–Zilber reduction $C_*(X\times Y){\xRightarrow{\ \ }}C_*X\otimes C_*Y$ is functorial, which implies that it restricts to a reduction $$C_*\big((A\times Y)\cup(X\times B)\big){\xRightarrow{\ \ }}(C_*A\otimes C_*Y)+(C_*X\otimes C_*B){\stackrel{\mathrm{def}}{=}}D_*.$$ The strong equivalences $C_*X{\xLeftrightarrow{\ \ \ }}C_*^{\mathrm{ef}}X$ and $C_*Y{\xLeftrightarrow{\ \ \ }}C_*^{\mathrm{ef}}Y$ induce a strong equivalence (by [@SergerGenova Proposition 61], whose construction is functorial, and hence applicable to the equivariant setting) $$C_*X\otimes C_*Y{\xLeftrightarrow{\ \ \ }}C_*^{\mathrm{ef}}X\otimes C_*^{\mathrm{ef}}Y$$ that, again, restricts to a strong equivalence of the subcomplex $D_*$ above with its obvious effective version $D_*^{\mathrm{ef}}$. The composition of these two strong equivalences finally yields 