Let $F,G:C\to D$ be functors. A natural transformation$\alpha:F\to G$ is equifibered (also called cartesian) if for any morphism$f:x\to y$ in $C$, the naturality square

$\array{ F x & \overset{F f}{\to} & F y\\
^{\alpha_x}\downarrow & & \downarrow^{\alpha_y} \\
G x & \underset{G f}{\to} & G y}$

The name “equifibered” comes from the fact that since $\alpha_x$ is a pullback of $\alpha_y$, they must have isomorphicfibers. (Of course, if $C$ is not connected, then being equifibered does not imply that all components of $\alpha$ have isomorphic fibers.)

There is an evident generalization to natural transformations between higher categories.

Properties

Given a functor $G:C\to D$, if $C$ has a terminal object$1$, then to give a functor $F$ and an equifibered transformation $F\to G$ is equivalent to giving a single object $F1$ and a morphism $F1 \to G1$. The rest of $F$ can then be constructed uniquely by taking pullbacks. This construction is important in the theory of clubs.