Family Fibration

Let $\mathcal{C}$ be a category. The family fibration $\mathrm{Fam}(\mathcal{C})\to\mathrm{Sets}$ has as objects families $I\to\mathcal{C}_{0}$, and morphisms $\mathrm{Fam}(\mathcal{C})_{u}(X_{i},Y_{j})$ over $u:I\to J$ families of morphisms $\mathcal{C}(X_{i},Y_{u(i)})$.

Cartesian Morphisms of the Family Fibration

The Cartesian Morphisms of the family fibration are the families of morphisms $f_{i}:\mathcal{C}(X_{i},Y_{u(i)})$ such that each $f_{i}$ is an isomorphism.