Lemma 99.3.2. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $f$ is limit preserving (Definition 99.3.1), then $f$ is limit preserving on objects (Criteria for Representability, Section 94.5).

**Proof.**
If for every directed limit $U = \mathop{\mathrm{lim}}\nolimits U_ i$ of affine schemes over $U$, the functor

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow (\mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i}) \times _{\mathcal{Y}_ U} \mathcal{X}_ U \]

is essentially surjective, then $f$ is limit preserving on objects. $\square$

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