Fibred category

Remarks

The cartesian morphism $\varphi$ over $f$ is unique up to a unique isomorphism over $\id_b$, and $x$ is often written $f^* y$. The fibre over $b$ is the subcategory of objects over $b$ and morphisms over $\id_b$. Fibrations over $\mathcal{B}$ correspond to pseudofunctors from $\mathcal{B}^{\mathrm{op}}$ to categories, by the Grothendieck construction.

Example: let $\mathcal{B}$ have pullbacks, let $\mathcal{E}$ be the arrow category whose objects are morphisms $e \colon a \to b$ of $\mathcal{B}$, and let $p$ send $e$ to its codomain $b$. Given $e \colon a \to b$ and $f \colon b' \to b$, the pullback square of $e$ along $f$ is a morphism of $\mathcal{E}$ over $f$. The universal property of the pullback says exactly that it is cartesian, so $p$ is a fibration.

Non-example: let $\mathcal{B}$ be the category with objects $0, 1$ and one non-identity arrow $f \colon 0 \to 1$. Let $\mathcal{E}$ have one object $y$ and only its identity, and let $p(y) = 1$. No object of $\mathcal{E}$ lies over $0$, so there is no morphism over $f$ ending at $y$, and $p$ is not a fibration.