Consider $C, A_1, A_2$ complete lattices and assume they are related by two Galois connections $\alpha_i : C \to A_i$, $\gamma_i: A_i \to C$, for $i\in \{1,2\}$.
I would like to construct a Galois connection $\alpha : C \to A_1 \times A_2$, $\gamma: A_1 \times A_2 \to C$ and prove it is such. Recall that in $A_1 \times A_2$ the order is taken pointwise.
I defined $\alpha(c) = (\alpha_1(c),\alpha_2(c))$ and $\gamma(a_1,a_2)=\gamma_1(a_1) \sqcap \gamma_2(a_2)$, where $\sqcap$ denotes the infimum in the complete lattice $C$. I proved it holds the adjunction property $\alpha(c) \le a \Leftrightarrow c \le \gamma(a)$, for every $c \in C, a \in A_1 \times A_2$. This property, by a proposition, implies the continuity of $\alpha$ and the monotonicity of $\gamma$, so it remains to prove $\gamma$ is continuous, i.e. for every $D \subseteq A_1\times A_2$ directed $$\gamma(\sqcup D) = \sqcup \gamma(D).$$
My problem is proving this equality. I tried writing $D = \{(a_i^1, a_i^2)\}_{i\in I}$, using monotonicity of $\gamma$ and continuity of $\gamma_i$ and now my thesis is $$ \left(\bigsqcup_{i \in I} \gamma_1(a_i^1)\right) \sqcap \left(\bigsqcup_{i \in I}\gamma_2(a_i^2)\right) \le \bigsqcup_{i \in I} (\gamma_1(a_i^1) \sqcap \gamma_2(a_i^2)).$$
I am stuked here.