I just learned about Kleene’s recursion theorem; the one that states that for any computable $Q$ there is an $e$ such that $\varphi_e(x)\simeq Q(e,x)$. Applying this to a Turing machine that halts if the binary relation $f(e)=x$ is met for $f$ computable, you can get there is an $e$ such that $W_e=\{f(e)\}$.
What I am wondering is if this can be extended to two sets? That is, given computable $f,g$ is there $a,b$ such that $W_a=\{f(a,b)\}$ and $W_b=\{g(a,b)\}$? I am aware of the recursion theorem with parameters and its corollary that there is some $a,b$ such that $\varphi_{f(a,b)}\simeq\varphi_a$ and $\varphi_{g(a,b)}\simeq\varphi_b$, but I don’t know if this is the thoerem I need since this seems to require making sure that $W_{f(a,b)}=\{f(a,b)\}$ for any computable $f$.
Eventually I want to see if this generalises to any number of sets that is: if $f_0,\cdots,f_k$ is a collection of computable functions then there exists $a_0,\cdots,a_k$ such that $W_{a_i}=\{f_i(a_0,\cdots,a_k)\}$ for each $i\leq k$, but I am stuck at the two case. Any help would be appreciated!
