I am reading the expository paper here. In particular, I am trying to understand the following proof: Let $\mathcal{C}$ be a category admitting all small coproducts. Let $\Sigma$ be a set of morphisms in $\mathcal{C}$ which admits a calculus of left fractions. If the set $\Sigma$ is closed under taking coproducts of its elements, then the localized category $\mathcal{C}[\Sigma^{-1}]$ admits small coproducts. Moreover, the quotient functor $\mathcal{C} \rightarrow \mathcal{C}[\Sigma^{-1}]$ 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, $$ \text{Hom}_{\mathcal{C}[\Sigma^{-1}]} \left( \coprod X_{i}, Y \right) \longrightarrow \prod \text{Hom}_{\mathcal{C}[\Sigma^{-1}]} (X_{i}, Y) $$ 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 $\bar{\Sigma}$ 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 $\mathcal{C}[\Sigma^{-1}]$. 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.


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