Question

Let $C$ be a complete category, and let $P$ be presheaf on $C$. Let $X\to P$ and $Z\to P$ be objects of $\mathcal{Y}\downarrow P$, where $\mathcal{Y}$ is the Yoneda embedding. Is the pullback $X\times_P Y$ in $Psh(C)$ representable? This is obvious when $P$ is representable, but I am not sure if it's true otherwise.

If it is true, does it still hold if $C$ is only finitely complete?

Answer 1

No. Take $C$ to be the terminal category, so that the category of presheaves is just $Set$. There is just one representable: the terminal set $1$. Let $P$ be a 2-element set, with elements $a: 1 \to P$ and $b: 1 \to P$. Then the pullback of these two morphisms is the empty set, which is not representable.