**Background:**
Theorem 6.8 in Suslin and Voevodsky's article "Singular homology of abstract algebraic varieties" states that there is an isomorphism between effective relative zero cycles $z_0^c(Z)^{eff}(S)$ and the monoid $Hom(S,\coprod_{d=0}^{\infty} Sym^d(Z))[1/p]$, where $Z$ is any algebraic scheme and $S$ is a normal connected algebraic scheme.

**Constructing the isomorphisms:**
The map $z_0^c(Z)^eff(S) \to Hom(S,\coprod_{d=0}^{\infty} Sym^d(Z))[1/p]$ is given as follows:
Suppose that $X \to Z \times S$ is a closed embedding such that the composition $X \to Z \times S \to S$ is finite and surjective. Then it is proved in the paper that we have a composition of morphisms $$f_X: = (S \to Sym^d(X/S) \to Sym^d(Z) \times S \to Sym^d(Z))$$ where the first morphism $S \to Sym^d(X/S)$ is a section of the projection $Sym^d(X/S) \to S$. Now the map $$z_0^c(Z)^{eff}(S) \to Hom(S,\coprod_{d=0}^{\infty} Sym^d(Z))[1/p]$$ is given by $$X \mapsto f_X.$$ The inverse map is given as follows: Consider the presheaf $$T \mapsto \mathbb{Z}[1/p](Hom_{Sch/k}(T,Z))$$ and let $\mathbb{Z}[1/p]_{qfh}(Z)$ denote the sheafification with respect to the qfh-topology. It turns out that the sheaf $\mathbb{Z}[1/p]_{qfh}(Z)$ is isomorphic to the sheaf of relative zero cycles $z_0^c(Z)$. Letting $q: Z^d \to Sym^d(Z)$ denote the projection, there is is a unique element $$u_d \in \mathbb{Z}[1/p]_{qfh}(Z)(Sym^d(Z))$$ such that $q^{*}(u_d) = \sum_i \pi_i \in \mathbb{Z}[1/p](Z)(Z^d)$ where $\pi_i: Z^d \to Z$ denotes the projection onto the i'th factor. Now given any $$f: S \to Sym^d(Z)$$ we map this to the element $$f^{*}(u_d) \in \mathbb{Z}[1/p]_{qfh}(Z)(S) = z_0^c(Z)(S)$$.
The article now states that one verifies easily that $f^{*}(u_d)$ is an effective relative zero cycle and that the resulting map
$$
Hom(S,\coprod_{d=0}^{\infty}Sym^d(Z))[1/p] \to z_0^c(Z)^{eff}(S)
$$
is an inverse to the one constructed before.

**My question:** How does one verify that these two maps constructed above are infact inverses to each other?

**What I have attempted:** I have tried showing that if $X \to Z \times S$ is a relative zero cycle then $f_X^{*}(u_d) = X$. Which using that $S \to Sym^d(X/S)$ is a section should be equivalent to showing that the class of $$\sum_i \pi_i|_{X^d/S \times_S Y} \in \mathbb{Z}[1/p]_{qfh}(Z)(X^d/S \times_S Y)$$ coincides with the class of $$\sum_{r \in Hom_S(Y,X)} p_1|_{X} \circ r|_{X^d/S \times_S Y} \in \mathbb{Z}[1/p]_{qfh}(Z)(X^d/S \times_S Y)$$ where $Y \to S$ is a normalization of $S$ in a field extension of $k(S)$ containing $k(X)$ and $p_1|_{X}$ is the composition $X \to Z \times S \to Z$. I cannot see why this should be the case though, and I have no idea how to prove that the opposite composition yields identity.