English
Let C be a cocomplete category whose opposite is locally small and well-powered, and suppose there exists a small separating set 𝒢 of objects in C. Then C has a terminal object (hence is complete).
Русский
Пусть C — кокомплетная категория, противолежащая к ней категория Cᵒᵖ локально малы и хорошо м domínio; пусть существует малый сепараторный набор 𝒢 объектов. Тогда в C существует терминальный объект (то есть C полно).
LaTeX
$$$$\text{HasColimits}(C) \land \text{WellPowered}(C^{op}) \land \text{LocallySmall}(C^{op}) \land \text{Small}(\mathcal{G}) \land \text{IsSeparating}(\mathcal{G}) \Rightarrow \text{HasTerminal}(C).$$$$
Lean4
/-- An ingredient of the proof of the Special Adjoint Functor Theorem: a cocomplete well-copowered
category with a small separating set has a terminal object.
In fact, it follows from the Special Adjoint Functor Theorem that `C` is already complete, see
`hasLimits_of_hasColimits_of_isSeparating`. -/
theorem hasTerminal_of_isSeparating [LocallySmall.{w} Cᵒᵖ] [WellPowered.{w} Cᵒᵖ] [HasColimitsOfSize.{w, w} C]
{𝒢 : Set C} [Small.{w} 𝒢] (h𝒢 : IsSeparating 𝒢) : HasTerminal C :=
by
haveI : Small.{w} 𝒢.op := small_of_injective (Set.opEquiv_self 𝒢).injective
haveI : HasInitial Cᵒᵖ := hasInitial_of_isCoseparating ((isCoseparating_op_iff _).2 h𝒢)
exact hasTerminal_of_hasInitial_op
