Let $U: \mathscr{A} \to \mathscr{C}$. If $\mathscr{E}$  has (enought large)  limits resp. colimits then the functor  $U^*: CAT(\mathscr{C}^{op}, \mathscr{E} ) \to  CAT(\mathscr{A}^{op}, \mathscr{E} ): P \mapsto   P\circ U^{op}$   has a left adjoin $U_!=Lan_{ U^{op}}$ (puntual Kan extention) resp. a right adjoint  $U_*=Ran_{ U^{op}  }$ and $U_! \dashv U^* \dashv U_*$ with  $U_!(P)(X):= {\underrightarrow{lim}} (P\circ \pi^{op}: (X\downarrow U)^{op}\to  \mathscr{A} ^{op}\to \mathscr{E})$

$U_*(P)(X):={\underleftarrow{lim}} (P\circ \pi^{op}: (U\downarrow X)^{op}\to  \mathscr{E} $ ). 

Let  $\mathscr{E} =Set$ and $\mathscr{A}, \mathscr{B}$ small (we can have more general conditions for the existence of puntual Kan extentions) ,  we have 

$U_!(P)$= $Lan_{ h_-} (h_U)(P)$ = 

$\underrightarrow{lim}$$_{(A, a)\in \mathscr{A} \downarrow P }$ $h_{ U(A)}$, 



$U_*(P)(X)  =\mathscr{A} ^>(h^U_X, P)$ 

indeed:  

$(\underrightarrow{lim}$$_{(A, a)$ $h_{U(A)}, Q)\cong$, 

$ {\underleftarrow{lim}}_{(A, a)}  QU(A)\cong $

$({\underrightarrow{lim}}_{(A,a)} h_A, Q\circ U)$ ;  

$(Q\circ U, P) \cong  ({\underrightarrow{lim}}_{(X, x)\in \mathscr{C}\downarrow Q } h^U_X , P) \cong  {\underleftarrow{lim}}_{(X, x)}  (h_X, \mathscr{A} ^>(h^U_-, P)) \cong   (Q(-), \mathscr{A}^>(h^U_-, P))$.


Then (answere for you): $U_∗(h_Y)(X)=(h^U_X,h_Y)$.