The Stacks project

Proof. If $A$ is an object of $\mathcal{C}_\Lambda $, then an object of the fiber category of $\mathcal{H} \times _\mathcal {F} \mathcal{G}$ over $A$ is a triple $(u, v, a)$ where $u \in \mathcal{H}(A)$, $v \in \mathcal{G}(A)$, and $a : f(u) \to g(v)$ is a morphism in $\mathcal{F}(A)$. Consider a diagram in $\mathcal{H} \times _\mathcal {F} \mathcal{G}$

in $\mathcal{C}_\Lambda $ with $A_2 \to A$ surjective. Since $\mathcal{H}$ and $\mathcal{G}$ satisfy (RS), there are fiber products $u_1 \times _ u u_2$ and $v_1 \times _ v v_2$ lying over $A_1 \times _ A A_2$. Since $\mathcal{F}$ satisfies (RS), Lemma 90.16.2 shows

are both fiber squares in $\mathcal{F}$. Thus we can view $a_1 \times _ a a_2$ as a morphism from $f(u_1 \times _ u u_2)$ to $g(v_1 \times _ v v_2)$ over $A_1 \times _ A A_2$. It follows that

is a fiber square in $\mathcal{H} \times _\mathcal {F} \mathcal{G}$ as desired. $\square$