The Stacks project

Proof. It follows from Lemmas 87.9.3 and 87.9.4 that an affine formal algebraic space satisfies (1) and (2). In order to prove the converse we may assume $X$ is not empty. Let $\Lambda $ be the category of representable morphisms $T \to X$ which are thickenings where $T$ is an affine scheme over $S$. This category is directed. Since $X$ is not empty, $\Lambda $ contains at least one object. If $T \to X$ and $T' \to X$ are in $\Lambda $, then we can factor $T \amalg T' \to X$ through $T'' \to X$ in $\Lambda $. Between any two objects of $\Lambda $ there is a unique arrow or none. Thus $\Lambda $ is a directed set and by assumption $X = \mathop{\mathrm{colim}}\nolimits _{T \to X\text{ in }\Lambda } T$. To finish the proof we need to show that any arrow $T \to T'$ in $\Lambda $ is a thickening. This is true because $T' \to X$ is a monomorphism of sheaves, so that $T = T \times _{T'} T' = T \times _ X T'$ and hence the morphism $T \to T'$ equals the projection $T \times _ X T' \to T'$ which is a thickening because $T \to X$ is a thickening. $\square$