The Stacks project

Lemma 4.18.3. Let $\mathcal{C}$ be a category. The following are equivalent:

  1. Nonempty finite limits exist in $\mathcal{C}$.

  2. Products of pairs and equalizers exist in $\mathcal{C}$.

  3. Products of pairs and fibre products exist in $\mathcal{C}$.

Proof. Since products of pairs, fibre products, and equalizers are limits with nonempty index categories we see that (1) implies both (2) and (3). Assume (2). Then finite nonempty products and equalizers exist. Hence by Lemma 4.14.10 we see that finite nonempty limits exist, i.e., (1) holds. Assume (3). If $a, b : A \to B$ are morphisms of $\mathcal{C}$, then the equalizer of $a, b$ is

\[ (A \times _{a, B, b} A)\times _{(pr_1, pr_2), A \times A, \Delta } A. \]

Thus (3) implies (2), and the lemma is proved. $\square$


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