Lemma 4.19.9. Let $\mathcal{I}$ be an index category satisfying the hypotheses of Lemma 4.19.8 above. Then colimits over $\mathcal{I}$ commute with fibre products and equalizers in sets (and more generally with finite connected limits).

Proof. By Lemma 4.19.8 we may write $\mathcal{I} = \coprod \mathcal{I}_ j$ with each $\mathcal{I}_ j$ filtered. By Lemma 4.19.2 we see that colimits of $\mathcal{I}_ j$ commute with equalizers and fibre products. Thus it suffices to show that equalizers and fibre products commute with coproducts in the category of sets (including empty coproducts). In other words, given a set $J$ and sets $A_ j, B_ j, C_ j$ and set maps $A_ j \to B_ j$, $C_ j \to B_ j$ for $j \in J$ we have to show that

$(\coprod \nolimits _{j \in J} A_ j) \times _{(\coprod \nolimits _{j \in J} B_ j)} (\coprod \nolimits _{j \in J} C_ j) = \coprod \nolimits _{j \in J} A_ j \times _{B_ j} C_ j$

and given $a_ j, a'_ j : A_ j \to B_ j$ that

$\text{Equalizer}( \coprod \nolimits _{j \in J} a_ j, \coprod \nolimits _{j \in J} a'_ j) = \coprod \nolimits _{j \in J} \text{Equalizer}(a_ j, a'_ j)$

This is true even if $J = \emptyset$. Details omitted. $\square$

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