I know that the question could be posed for any presheaf category, but simplicial sets are the only one having a name for natural transformations from representables, so I'll stick with that. So, go in the category of simplicial set and consider a pullback of this form:
$\require{AMScd}$ \begin{CD} P @>>> X\\ @V V V @VV f V\\ \Delta^n @>>\sigma> Y \end{CD} Obviously, since limits are computed levelwise, we already have a concrete characterization of $P$. But, since by Yoneda Lemma $\sigma$ is completely defined by an element $y=\sigma_n(id_n)\in Y_n$, I would really like $P$ to be in some way identified with (or at least completely determined by) the fiber of $y$ along $f$, but I am failing all my attempts. Is my desire without hope? Am I failing to see something obvious?
An $n$-simplex of $P$ over the non degenerate $n$-simplex of $\Delta^n$ is by definition given by an $n$-simplex in the fiber of $f$ over $y.$ Analogously the simplices of $P$ over any simplex of $\Delta^n$ are the simplices of $X$ over the image of that simplex in $Y,$ which is always some possibly-degenerated face of $y.$ In other words, $P$ is indeed almost exactly the “fiber over $y$” of $f.$ The only refinement to that picture is that if $\sigma$ is not injective, then $P$ will not be a subobject of $X,$ but will instead duplicate simplices of $X$ mapping to simplices in $Y$ that are hit by multiple simplices in $\Delta^n.$ It should be informative to calculate the case where $f=\sigma$ is the map identifying the two $0$-simplices of $\Delta^1.$