Suppose $F: C \rightarrow D$ is the left adjoint to a functor $G$. Then is it true that the functor $F^{\star}:[C : Sets]$ defined by prescomposing a functor $P: C \rightarrow Sets$ is still left adjoint to the functor $G^{\star}$ defined similarly? 


Or would the adjunction be flipped an $F^{\star}$ becomes right adjoint to $G^{\star}$?



Induced adjunctions
Yes,


Let $Q \in [\mathfrak{D}:Sets]$,$A \in \mathfrak{D}$ $P \in [\mathfrak{C}:Sets]$ and $B \in \mathfrak{C}$. 


If $F: \mathfrak{C} \rightarrow \mathfrak{D}$ is left adjoint to $G$, then there are unit and counit natural transformations $\eta: F \circ G \rightarrow 1\_{\mathfrak{D}}$ and $\epsilon: G\circ F \rightarrow 1\_{\mathfrak{C}}$ respectively. 


These induce natural ismorphisms $\epsilon^{\star}: G^{\star}\circ F^{\star} (Q)(A) = Q ((F\circ G) (A)) \rightarrow Q(1\_{\mathfrak{D}}(A)) = Q(A) \cong 1\_{[\mathfrak{D}:Sets]} (Q(A))$. Therefore, the counit has induces a unit. 


Dually, we have $\eta^{\star}: F^{\star}\circ G^{\star} (P)(B) = P ((G\circ F) (B)) \rightarrow P(1\_{\mathfrak{C}}(B)) = P(B) \cong 1\_{[\mathfrak{C}:Sets]} (P(B))$ and similarly, the unit has induced a counit.


Therefore if $F$ is left adjoint to $G$, then $G^{\star}$ is left adjoint to $F^{\star}$. 


