Kerodon

Exercise 7.6.5.24. In the situation of Proposition 7.6.5.23, suppose that $X_0$ and $X_1$ admit a fiber product over $X$. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves the product of $X_0$ and $X_1$ (that is, $F( \pi _0 )$ and $F( \pi _1 )$ exhibit $F( X_0 \times X_1 )$ as a product of $F(X_0)$ with $F(X_1)$ in the $\infty $-category $\operatorname{\mathcal{D}}$). Show that $F$ preserves the fiber product of $X_0$ with $X_1$ over $X$ if and only if it preserves the equalizer of the morphisms $g_0$ and $g_1$.