proposition: Lemma coeq_hyp_eq_hyp : forall a u v y, Category.axioms a -> coequalizer_hypothesis a u v y = equalizer_hypothesis (opp a) (flip u) (flip v) (flip y).
proof: Proof. ir. ap iff_eq; ir. uh H0; uhg; ee. uh H0; uhg; ee. rww mor_opp. rww flip_flip. rww mor_opp. rww flip_flip. rw source_flip. rw source_flip. am. alike. alike. rw target_flip; try alike. rww target_flip; try alike. rww mor_opp; rww flip_flip. rww target_flip; try alike. rww source_flip; try alike. sy; am. uh H0; ee; alike. rw comp_opp. rw comp_opp. rw flip_flip. rw flip_flip. rw flip_flip. ap uneq. am. rww mor_opp; rww flip_flip. uh H0; ee; am. rww mor_opp; rww flip_flip. rww target_flip; try alike. rww source_flip; try alike. uh H0; ee. wr H6. sy; am. uh H0; ee; alike. rww mor_opp; rww flip_flip. uh H0; ee; am. rww mor_opp; rww flip_flip. rww target_flip; try alike. rww source_flip; try alike. sy; am. uh H0; ee; alike. uh H0; ee. uh H0; ee. uhg; dj. uhg; dj. rwi mor_opp H0. rwi flip_flip H0. am. rwi mor_opp H4. rwi flip_flip H4. am. wr target_flip; try alike. rw H6. rw target_flip; try alike. tv. wr source_flip; try alike. rw H5. rww source_flip; try alike. rwi mor_opp H1. rwi flip_flip H1. am. uh H7; ee. wr target_flip; try alike. wr H2. rw H5. rww source_flip; try alike. sy; am. rwi comp_opp H3. rwi flip_flip H3. rwi flip_flip H3. transitivity (flip (flip (comp a y u))). rww flip_flip. rw H3. rw comp_opp. rw flip_flip. rw flip_flip. rw flip_flip. tv. am. am. wr H5. am. am. am. am. Qed.
imports: Arith colimits little_cat fc_limits cardinal