Vtyjkyuk

I am reading the expository paper here. In particular, I am trying to understand the following proof: Let $mathcalC$ be a category admitting all small coproducts. Let $Sigma$ be a set of morphisms in $mathcalC$ which admits a calculus of left fractions. If the set $Sigma$ is closed under taking coproducts of its elements, then the localized category $mathcalC[Sigma^-1]$ admits small coproducts. Moreover, the quotient functor $mathcalC rightarrow mathcalC[Sigma^-1]$ preserves small coproducts. This is Proposition 3.5.1 on page 12.

To prove this, the author sets out to show that the canonical morphism,
$$
textHom_mathcalC[Sigma^-1] left( coprod X_i, Y right) longrightarrow prod textHom_mathcalC[Sigma^-1] (X_i, Y)
$$
is a bijection of sets for all objects $Y$.

The statement is straightfoward enough. But I am stuck with a particular claim he makes on page 13. About halfway down the page he constructs the morphisms $beta_i: Z rightarrow Z_i$, and claims that each $beta_i$ belongs to the saturation $barSigma$ of $Sigma$. I don't see why this is true at all. He defines the saturation to be the family of morphisms which become isomorphisms in the localization $mathcalC[Sigma^-1]$. He notes that a morphism $phi$ is in the saturation of $Sigma$ if and only if there are morphisms $phi'$ and $phi''$ such that $phi circ phi'$ and $phi'' circ phi$ are both in $Sigma$. It is obvious that the $phi'$ for $beta$ in his construction is $sigma$, but what $phi''$ would work? I don't see why $beta_i$ is in the saturation for $Sigma$ at all.