Does the composition of adjoint functors again form an adjunction?
Say $\langle F_1,G^1\rangle$ is an adjunct pair between two categories A and B and $\langle F_2,G^2\rangle$ is also an adjoint pair between the categories B and C, then would $\langle F_2\circ F_1,G^1\circ G^2\rangle$ be an adjoint pair between the categories A and C?
If not is there some way of obtain an adjunction given the assumptions made on the 4 functors, or by adding extra properties to the categories?
Particularly, here I am assuming A is the category $_RMod$ for some (apriori) arbitrary ring.
Best Answer
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 $G^1\circ G^2$.