Let $A:= \{\langle i,j\rangle \in \mathbb{N} \mid W_i \cap W_j = \emptyset\}$ where $W_i$ corresponds to the domain of the universal partial computable function $\varphi_i$.
Let us show that $A \in \Pi_2^0$.
$z=\langle i,j\rangle \in A \Leftrightarrow W_i \cap W_j = \emptyset \Leftrightarrow \forall x \exists t \ (T[\pi_1(i),x,\pi_1(t)] \rightarrow \neg T[\pi_2(j),x,\pi_2(t)]) \Leftrightarrow \forall x \exists t \ (\neg T[\pi_1(i),x,\pi_1(t)] \vee \neg T[\pi_2(j),x,\pi_2(t)])$
where the predicate : $\neg T[\pi_1(i),x,\pi_1(t)] \vee \neg T[\pi_2(j),x,\pi_2(t)] \in \Delta_0$ ($T$ stands for the Kleene's predicate). Hence $A \in \Pi_2^0$.
Let us show that $A$ is $\Pi_2^0$-complete. We will prove that $\text{Tot}\le_m A$ where $\text{Tot}:= \{e \in \mathbb{N} \mid W_e = \mathbb{N}\}$ (indeed $\text{Tot}$ is $\Pi_2^0$-complete).
Let $e \in \text{Tot}$ hence $W_e = \mathbb{N}$. Moreover, we can notice that if $j \in \mathbb{N}$ is the index of the nowhere defined function (which is computable) then $W_e \cap W_j= \emptyset$. Hence if we consider the pair $\langle e,j \rangle$ (that encoding is in fact primitive recursive), we get that $\langle e,j \rangle \in A$.
Hence $e \in \text{Tot} \Leftrightarrow \langle e,j \rangle \in A$ and then $\text{Tot}\le_m A$. We can conclude that $A$ is $\Pi_2^0$-complete.
I was wondering if the last point of the proof was correct and if I should write more carefully the total computable function used in the reduction step ?
Also, is there a more "algorithmic" way (with machines for example) to see the reduction step ?
Thanks in advance !