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Let $S$ be a small family of arrows in a locally presentable category $\mathcal{K}$.

It is known that the category $\mathcal{K}[S^{-1}]$ is reflective in $\mathcal{K}$ and correspond to the solution of the orthogonality problem associated to $S$.

Can I infer that a map is an iso in $\mathcal{K}[S^{-1}]$ if and only if it comes from a map in $S$?

This question is strongly related to this other one. In that case, the localization might not be reflective, and this difference might be quite relevant in the answer.

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No, you cannot infer this. The class of morphisms inverted by any functor must satisfy the two-for-three, even the two-for-six, property. But $S$ is completely arbitrary. Consider, for a dramatic example, $\mathrm{Set}[(\emptyset \to \{*\})^{-1}]$, where inverting a single arrow is equivalent to inverting all arrows. Furthermore, there are many more conditions than just two-for-$n$ in this situation. Indeed, the class of morphisms inverted by any left adjoint is closed under colimits in the arrow category.

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