Cartesian Morphism

Definition

Let $\mathcal{E},\mathcal{B}$ be categories.
Let $p:\mathcal{E}\Rightarrow \mathcal{B}$ be a functor.
A morphism $\alpha :\mathcal{E}[X,Y]$ is Cartesian over $f:\mathcal{B}[pX,pY]$ iff

$$\begin{array}{rl}& \forall g:\mathcal{E}[Z,Y]\\ & \forall h:\mathcal{B}[pZ,pX]\\ & pg=f\circ h⟹\\ & \exists !\tilde{g}:\mathcal{E}[Z,X]\\ & g=\alpha \circ \tilde{g}\wedge p\tilde{g}=h\end{array}$$

See also

Used as part of Grothendieck Fibration.