proposition: Lemma terminal_in_full_subcategory {C : category} (C' : hsubtype (ob C)) (TC : Terminal C) (TC' : C' (TerminalObject TC)) : Terminal (full_sub_precategory C').
proof: Proof. use tpair. - use precategory_object_in_subcat. + exact (TerminalObject TC). + assumption. - cbn. intros X. use make_iscontr. + use morphism_in_full_subcat. apply TerminalArrow. + intro; apply eq_in_sub_precategory; cbn. apply TerminalArrowUnique. Defined.
imports: UniMath.Foundations.PartA UniMath.Foundations.Sets UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Subcategory.Core UniMath.CategoryTheory.Subcategory.Full UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Graphs.Colimits