Composition of adjoint functors
Let $a$ be an object in $A$, $c$ be an object in $C$ and $b$, $\tilde{b}$ be objects in $B$.
The adjunctions hypothesized give:
$C(F_2(b),c)\cong B(b,G^2(c))$ and $B(F_1(a),\tilde{b}) \cong A(a,G^1(\tilde{b}))$.
Now setting $b:=F_1(a)$ and $\tilde{b}:=G^2(c)$ it may be concluded:
$A(a,G^1(G^2(c))) \cong B(F_1(a),G^2(c)) \cong C(F_2(F_1(a)),c)$.
So $F_2\circ F_1$ is left adjoint to $G^1\circ G^2$.