TITLE: Ejemplo de una fibra de una categoría
QUESTION [5 upvotes]: Sea el funtor $P_{\mathcal{F}}: \mathcal{F}\longrightarrow{\mathcal{C}}$.
Dado un objeto $U$ de $\mathcal{C}$, la fibra de $\mathcal{F}$ sobre $U$ denotada por $\mathcal{F}(U)$ es la subcategoría de $\mathcal{F}$ cuyos objetos son los objetos $\xi$ de $\mathcal{F}$ con $P_\mathcal{F}\xi=U$ y cuyas flechas son flechas $\phi$ en $\mathcal{F}$ con $P_\mathcal{F}\phi=id_U$.
Estoy queriendo un ejemplo en donde tengamos dos objetos $U$ y $V$ en $\mathcal{C}$ que son isomorfos pero tal que $\mathcal{F}(U)$ es vacío pero $\mathcal{F}(V)$  no lo es.  Muchas gracias.

English translation:
Let $P_\mathcal{F}:\mathcal{F}\to\mathcal{C}$ be a functor.  Given an object $U$ of $\mathcal{C}$, the fiber of $\mathcal{F}$ over $U$, denoted by $\mathcal{F}(U)$, is the subcategory of $\mathcal{F}$ whose objects are the objects $\xi$ of $\mathcal{F}$ with $P_\mathcal{F}\xi=U$ and whose arrows are the arrows $\phi$ in $\mathcal{F}$ with $P_\mathcal{F}\phi=id_U$.  
I'm wanting an example in which we have two objects $U$ and $V$ in $\mathcal{C}$ which are isomorphic but such that $\mathcal{F}(U)$ is empty but $\mathcal{F}(V)$ is not.  Thank you very much.

REPLY [2 votes]: Let $\mathcal{C}$ be your favorite category with two objects $U$ and $V$ which are isomorphic but not equal.  Let $\mathcal{F}$ be the full subcategory of $\mathcal{C}$ whose only object is $V$, and let $P_\mathcal{F}:\mathcal{F}\to\mathcal{C}$ be the inclusion functor.  Then $\mathcal{F}(U)$ is empty but $\mathcal{F}(V)$ is not.
More generally, there's no reason to think that an isomorphism between $U$ and $V$ should be reflected in the fibers of $\mathcal{F}$, since that isomorphism may not be in the image of $\mathcal{F}$.