Lemma 4.8.1. Let $\mathcal{C}$ be a category. Let $F, G, H : \mathcal{C}^{opp} \to \textit{Sets}$ be functors. Let $a : F \to G$ and $b : H \to G$ be transformations of functors. Then the fibre product $F \times _{a, G, b} H$ in the category $\textit{PSh}(\mathcal{C})$ exists and is given by the formula
\[ (F \times _{a, G, b} H)(X) = F(X) \times _{a_ X, G(X), b_ X} H(X) \]
for any object $X$ of $\mathcal{C}$.
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