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 4.14.11 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

$(A \times _{a, B, b} A)\times _{(\text{pr}_1, \text{pr}_2), A \times A, \Delta } A.$

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

Comment #7476 by R.K on

Another description of equalizers as built from fiber products and products (which seems easier to me for it only involves one product and one fiber product) is $B \times_{(\textbf{1}_B,\textbf{1}_B), B\times B, (f,g)} \times A$.

Comment #7624 by on

Indeed! Hmm... not worth changing to me but if others chime in... also this occurs in more places than just here.

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