Lemma isunitindirprod (X Y : monoid) : isunit (@op (setwithbinopdirprod X Y)) (1 ,, 1). Proof. split. - intro xy. induction xy as [ x y ]. simpl. apply pathsdirprod. apply (lunax X). apply (lunax Y). - intro xy. induction xy as [ x y ]. simpl. apply pathsdirprod. apply (runax X). apply (runax Y). Qed.
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets.