Is there a yoneda lift of functors? The Yoneda embedding or lift of a category $A$ into its category of presheaves $\hat{A}:=Set^{A^{op}}$ is $y_A:A \rightarrow \hat{A}$
Hence for a catgory $B$, we also have $y_B:B \rightarrow \hat{B}$.
Given a functor $F:A \rightarrow B$ can we lift it to a functor $\hat{F}:\hat{A} \rightarrow \hat{B}$?
There is the obvious embedding $y_{B^A}:B^A \rightarrow \hat{B^A}$, and this would work if $\hat{B^A} \backsimeq \hat{B}^{\hat{A}} $; but I don't think is right.

You should assume that $A,B$ are small.
Then $F : A \to B$ lifts to a functor $\widehat{F} : \widehat{A} \to \widehat{B}$, namely the left Kan extension of $y_B F$ along $y_A$. Explicitly, we have
$$\widehat{F}(S) = \mathrm{colim}_{p \in S(a)} \hom(-,F(a))$$
More abstractly, we can view $\widehat{F}$ as the left adjoint of the obvious functor $F^* : \widehat{B} \to \widehat{A}$, $S \mapsto S \circ F^{\mathrm{op}}$.
Also notice that $y_A : A \hookrightarrow \widehat{A}$ is the free cocompletion of $A$ under colimits. The existence of $\widehat{F}$ is a special case of this universal property.
