Lemma 19.7.2. Suppose that $\mathcal{G}_ i$, $i\in I$ is set of abelian sheaves on $\mathcal{C}$. There exists an ordinal $\beta $ such that for any sheaf $\mathcal{F}$, any $i\in I$, and any map $\varphi : \mathcal{G}_ i \to J_\beta (\mathcal{F})$ there exists an $\alpha < \beta $ such that $ \varphi $ factors through $J_\alpha (\mathcal{F})$.
Proof. This reduces to the case of a single sheaf $\mathcal{G}$ by taking the direct sum of all the $\mathcal{G}_ i$.
Consider the sets
\[ S = \coprod \nolimits _{U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})} \mathcal{G}(U). \]
and
\[ T_\beta = \coprod \nolimits _{U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})} J_\beta (\mathcal{F})(U) \]
The transition maps between the sets $T_\beta $ are injective. If the cofinality of $\beta $ is large enough, then $T_\beta = \mathop{\mathrm{colim}}\nolimits _{\alpha < \beta } T_\alpha $, see Sites, Lemma 7.17.10. A morphism $\mathcal{G} \to J_\beta (\mathcal{F})$ factors through $J_\alpha (\mathcal{F})$ if and only if the associated map $S \to T_\beta $ factors through $T_\alpha $. By Sets, Lemma 3.7.1 if the cofinality of $\beta $ is bigger than the cardinality of $S$, then the result of the lemma is true. Hence the lemma follows from the fact that there are ordinals with arbitrarily large cofinality, see Sets, Proposition 3.7.2. $\square$
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