Lemma 10.127.4. Let $R \to \Lambda $ be a ring map. Let $\mathcal{E}$ be a set of $R$-algebras such that each $A \in \mathcal{E}$ is of finite presentation over $R$. Then the following two statements are equivalent

$\Lambda $ is a filtered colimit of elements of $\mathcal{E}$, and

for any $R$ algebra map $A \to \Lambda $ with $A$ of finite presentation over $R$ we can find a factorization $A \to B \to \Lambda $ with $B \in \mathcal{E}$.

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