Existence of the homotopy category functor h from simplicial sets to Cat is equivalent to specific weighted colimits in Cat, making the nerve embedding reflective and establishing that Cat is complete and cocomplete.
Borceux.Handbook of Categorical Algebra, volume 2 : Categories and Structures
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Revisiting colimits in $\mathbf{Cat}$ and homotopy category
Existence of the homotopy category functor h from simplicial sets to Cat is equivalent to specific weighted colimits in Cat, making the nerve embedding reflective and establishing that Cat is complete and cocomplete.