`∀[C:SmallCategory]. ∀[M:Monad(C)]. ∀[x,y:cat-ob(C)]. ∀[f:cat-arrow(C) x M(y)].`
`  (monad-extend(C;M;x;y;f) ∈ cat-arrow(C) M(x) M(y))`

Proof

Definitions occuring in Statement :  monad-extend: `monad-extend(C;M;x;y;f)` monad-fun: `M(x)` cat-monad: `Monad(C)` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` uall: `∀[x:A]. B[x]` member: `t ∈ T` apply: `f a`
Definitions unfolded in proof :  mk-functor: mk-functor functor-comp: `functor-comp(F;G)` functor-ob: `ob(F)` uimplies: `b supposing a` subtype_rel: `A ⊆r B` nat-trans: `nat-trans(C;D;F;G)` and: `P ∧ Q` pi2: `snd(t)` monad-op: `monad-op(M;x)` monad-extend: `monad-extend(C;M;x;y;f)` pi1: `fst(t)` monad-functor: `monad-functor(M)` monad-fun: `M(x)` spreadn: spread3 cat-monad: `Monad(C)` member: `t ∈ T` uall: `∀[x:A]. B[x]`
Lemmas referenced :  small-category_wf cat-monad_wf cat-ob_wf monad-fun_wf functor-comp_wf cat-arrow_wf subtype_rel-equal functor-arrow_wf functor-ob_wf cat-comp_wf
Rules used in proof :  independent_isectElimination because_Cache hypothesis hypothesisEquality isectElimination extract_by_obid introduction applyEquality sqequalRule productElimination rename thin setElimination sqequalHypSubstitution cut isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}[C:SmallCategory].  \mforall{}[M:Monad(C)].  \mforall{}[x,y:cat-ob(C)].  \mforall{}[f:cat-arrow(C)  x  M(y)].