Theorem. Family construction is free coproduct completion [mde-XCVI]

The family construction \(\textsf {Fam}(\mathbb {C})\) over a small category \(\mathbb {C}\) is the free (small) coproduct completion of \(\mathbb {C}\).

The idea is that a family \(f : I \to \mathbb {C}_{\text {obj}}\) is a formal coproduct \(\prod _I f(i)\).

Proof.  [#244]

Let \(\mathbb {D}\) be a category with coproducts and \(F : \mathbb {C} \to \mathbb {D}\) a functor. We show that there is a unique (up to natural isomorphism) coproduct preserving \(\hat {F} : \textsf {Fam}(\mathbb {C}) \to \mathbb {D}\) commuting with the embedding \(i : \mathbb {C} \to \textsf {Fam}(\mathbb {C})\) up to natural isomorphism -- this gives a left pseudoadjoint to the forgetful \(\mathbf {Cat_\times } \to \mathbf {Cat}\).

On objects, define \(\hat {F}\) by \(\{A_i\}_{i \in I} \mapsto \coprod _{i \in I} F(A_i)\) and on morphisms by \(\alpha \mapsto [F(\alpha _i)]\). Of course, this is defined only up to natural isomorphism, and it is easy to check that it commutes with the inclusion.