The total category of a displayed category over $\mathrm{𝒞︀}$ collects the displayed data into a single category. Its objects are pairs of an object $𝑥$ of $\mathrm{𝒞︀}$ with an object over it, and its morphisms are pairs of a morphism $𝑓:𝑥\to 𝑦$ with a displayed morphism over $𝑓$.

Projecting out the first components is a functor . Constructions presented displayed — algebras, Eilenberg–Moore categories — get their forgetful functor for free as this projection.