Definition. Family construction [mde-7LXB]

Given a category \(\mathbb {C}\), the category \(\text {Fam}(\mathbb {C})\) of families in \(\mathbb {C}\) has

1. Objects: families of objects in \(\mathbb {C}\), i.e. functions \(I \to \mathbb {C}_{\text {obj}}\), for indexing sets \(I\).
2. Morphisms: reindexings \(f : I \to J\) along with morphisms \(\alpha _i : A_i \to B_{f_i}\) in \(\mathbb {C}\), i.e. laxly commuting triangles,