I have the following statement:
Let $G:\mathcal B\to\mathcal A$ be a functor. Then $G$ has a left adjoint if and only if for each $A\in\mathcal A$, the category $(A\Rightarrow G)\ (A\downarrow G)$ has initial object.
Is this the dual of the statement above:
Let $F:\mathcal A\to\mathcal B$ be a functor. Then $F$ has right adjoint if and only if for each $B\in\mathcal B$ category $(F\Rightarrow B)\ (F\downarrow B)$ has terminal object?
Seems right to me: the correspondence sending $a\in\cal A$ in the first entry of the initial object $(x,\xi_a : a\to Gx)$ in $(a\downarrow G)$ is a functor (initiality of $\xi_-$'s defines it on arrows), and there is a bijection ${\cal A}(a, Gb)\cong {\cal B}(\text{init}(a\downarrow G), b)$ given by $(f : x\to b)\mapsto Gf\circ \xi_a$; equivalently, these $\xi_a$'s are precisely the unit components of the adjuction, and you can show the zig-zag identities.
Now, the second statement is perfectly dual: you get the components of the counit $\zeta : FG \Rightarrow 1$ after having defined the right adjoint $G$ to $F$ sending $b$ to the terminal object of $(F\downarrow b)$.