TITLE: Lifting adjunctions
QUESTION [5 upvotes]: Is there a convenient hom-set proof that an adjunction $ F \colon C \rightleftarrows D \colon G $ where $ F $ is left adjoint to $ G $ can be lifted to an adjunction of functor categories $ F_{*} \colon \operatorname{Funct}(J,C) \rightleftarrows \operatorname{Funct}(J,D) \colon G_{*} $ given by post-composition by $ F $ and $ G $, where $ J $ is any (small) category?
I can prove this by defining a unit and counit explicitly, but I'd like a hom-set proof.

REPLY [3 votes]: For $X : J \to C$, $Y : J \to D$ we have
$$\hom(F \circ X,Y) \cong \int_{j \in J} \hom(F(X(j)),Y(j)) \cong \int_{j \in J} \hom(X(j),G(Y(j)) \cong \hom(X,G \circ Y).$$
Here, $\int$ refers to an end. If you don't know this formalism, you might as well translate the proof as follows: A natural transformation $F \circ X \to Y$ consists of a family of maps $F(X(j)) \to Y(j)$, which are natural in $j$. Since $F$ is left adjoint to $G$, these correspond to maps $X(j) \to G(Y(j))$. Since the bijection in the definition of an adjunction is natural, the maps $X(j) \to G(Y(j))$ are also natural. So you end up with a natural transformation $X \to G \circ Y$.