# Composition of adjoint functors

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.

• Woops.. I recopied the wrong thing from my note pad :0
– ABIM
Commented Feb 19, 2014 at 14:29
• Ok. What have you tried? The proof is one line long. Commented Feb 19, 2014 at 14:37

## 1 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 $$G^1\circ G^2$$.

• I guess I should stop being afraid of my own solutions :P Thanks Martin :)
– ABIM
Commented Feb 19, 2014 at 14:52
• Also note: a composition of natural isomorphisms is still natural. Commented Jan 9, 2022 at 15:21