Certain results involving "higher structures" are not currently accessible to computer formalization because the prerequisite $\infty$-category theory has not been formalized. To support future work on formalizing $\infty$-category theory in Lean's mathematics library, we formalize some fundamental constructions involving the 1-category of categories. Specifically, we construct the left adjoint to the nerve embedding of categories into simplicial sets, defining the homotopy category functor. We prove further that this adjunction is reflective, which allows us to conclude that Cat has colimits. To our knowledge this is the first formalized proof that the nerve functor is a fully faithful right adjoint and that the category of categories is cocomplete.