Lemma 4.31.12. Let

be a $2$-fibre product of categories. Then there is a canonical isomorphism

Lemma 4.31.12. Let

\[ \xymatrix{ \mathcal{C}' \ar[r] \ar[d] & \mathcal{S} \ar[d]^\Delta \\ \mathcal{C} \ar[r]^{G_1 \times G_2} & \mathcal{S} \times \mathcal{S} } \]

be a $2$-fibre product of categories. Then there is a canonical isomorphism

\[ \mathcal{C}' \cong (\mathcal{C} \times _{G_1, \mathcal{S}, G_2} \mathcal{C}) \times _{(p, q), \mathcal{C} \times \mathcal{C}, \Delta } \mathcal{C}. \]

**Proof.**
An object of the right hand side is given by $((C_1, C_2, \phi ), C_3, \psi )$ where $\phi : G_1(C_1) \to G_2(C_2)$ is an isomorphism and $\psi = (\psi _1, \psi _2) : (C_1, C_2) \to (C_3, C_3)$ is an isomorphism. Hence we can associate to this the triple $(C_3, G_1(C_1), (G_1(\psi _1^{-1}), \phi ^{-1} \circ G_2(\psi _2^{-1})))$ which is an object of $\mathcal{C}'$. Details omitted.
$\square$

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