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Let $k$ be a field of characteristic zero. Fix a smooth algebraic curve $X$and a split reductive group $G$ over $k$. We show that the Beilinson--Drinfeldaffine Grassmannian $\mathrm{Gr}_{G, \mathrm{Ran}(X)}$ is the presheaf colimitof the reduced ind-schemes $(\mathrm{Gr}_{G, X^I})^{\mathrm{red}}$ for finitesets $I$. This implies that every map from an affine $k$-scheme to$\mathrm{Gr}_{G, \mathrm{Ran}(X)}$ factors through a reduced quasi-projective$k$-scheme. In the course of the proof, we generalize the notion of 'reductionof a scheme' to apply to any presheaf, and we show that this notion iswell-behaved on any pseudo-ind-scheme which admits a colimit presentation whoseindexing category satisfies the amalgamation property.

$\mathrm{Gr}_{G, \mathrm{Ran}(X)}$ is reduced