`F(g ∘ f) = F(g) ∘ F(f)`

+
+* if `η`

is a natural transformation from `F` to `G`, then for every `f:C1→C2`

in `F` and `G`'s source category `η[C2] ∘ F(f) = G(f) ∘ η[C1]`

.
Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas.
-Recall that join is a natural transformation from the (composite) functor MM to M. So for elements C1 in `f:C1→C2`

in + (1) join[C2] ∘ MM(f) = M(f) ∘ join[C1] ++ +Next, consider the composite transformation

`((join MG') -v- (MM γ))`

.
+
+* `γ`

is a transformation from `G` to `MG'`, and assigns elements `C1` in `γ\*: G(C1) → MG'(C1)`

. `(MM γ)`

is a transformation that instead assigns `C1` the morphism `MM(γ\*)`

.
+
+* `(join MG')` is a transformation from `MMMG'` to `MMG'` that assigns `C1` the morphism `join[MG'(C1)]`.
- (1) join[b] ∘ MM(f) = M(f) ∘ join[a]
+Composing them:
+
++ (2)-Next, consider the composite transformation ((join MG') -v- (MM γ)). - γ is a transformation from G to MG', and assigns elements C1 in`((join MG') -v- (MM γ))`

assigns to `C1` the morphism`join[MG'(C1)] ∘ MM(γ*)`

. +

`((M γ) -v- (join G))`

.
-Next, consider the composite transformation ((M γ) -v- (join G)).
+(3) This assigns to C1 the morphism M(γ*) ∘ join[G(C1)]. +-So for every element C1 of

((join MG') -v- (MM γ))[C1], by (2) is: join[MG'(C1)] ∘ MM(γ*), which by (1), with f=γ*: G(C1)→MG'(C1) is: M(γ*) ∘ join[G(C1)], which by 3 is: ((M γ) -v- (join G))[C1] ++ +So our **(lemma 1)** is: -So our (lemma 1) is: ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), where γ is a transformation from G to MG'. +

+ ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), where γ is a transformation from G to MG'. +-Next recall that unit is a natural transformation from 1C to M. So for elements C1 in

`f:a→b`

in (4) unit[b] ∘ f = M(f) ∘ unit[a] ++ +Next consider the composite transformation

`((M γ) -v- (unit G))`

:
+
++ (5) This assigns to C1 the morphism M(γ*) ∘ unit[G(C1)]. +-Next consider the composite transformation ((M γ) -v- (unit G)). (5) This assigns to C1 the morphism M(γ*) ∘ unit[G(C1)]. +Next consider the composite transformation

`((unit MG') -v- γ)`

.
-Next consider the composite transformation ((unit MG') -v- γ). (6) This assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
++ (6) This assigns to C1 the morphism unit[MG'(C1)] ∘ γ*. +So for every element C1 of

((M γ) -v- (unit G))[C1], by (5) = M(γ*) ∘ unit[G(C1)], which by (4), with f=γ*: G(C1)→MG'(C1) is: unit[MG'(C1)] ∘ γ*, which by (6) = ((unit MG') -v- γ)[C1] ++ +So our **(lemma 2)** is: -So our lemma (2) is: (((M γ) -v- (unit G)) = ((unit MG') -v- γ)), where γ is a transformation from G to MG'. +

+ (((M γ) -v- (unit G)) = ((unit MG') -v- γ)), where γ is a transformation from G to MG'. +-Finally, we substitute ((join G') -v- (M γ) -v- φ) for γ <=< φ in the monad laws. For simplicity, I'll omit the "-v-". +Finally, we substitute

`((join G') -v- (M γ) -v- φ)`

for `γ <=< φ`

in the monad laws. For simplicity, I'll omit the "-v-".
+for all φ,γ,ρ in T, where φ is a transformation from F to MF', γ is a transformation from G to MG', R is a transformation from R to MR', and F'=G and G'=R: (i) γ <=< φ etc are also in T @@ -379,10 +418,12 @@ Finally, we substitute ((join G') -v- (M γ) -v- φ) for γ <=< &ph which will in turn be true just in case: (iii.2') (join (unit M)) = the identity transformation +Collecting the results, our monad laws turn out in this format to be: + when φ a transformation from F to MF', γ a transformation from F' to MG', ρ a transformation from G' to MR' all in T: (i') ((join G') (M γ) φ) etc also in T @@ -392,6 +433,7 @@ Collecting the results, our monad laws turn out in this format to be: (iii.1') (join (M unit)) = the identity transformation (iii.2')(join (unit M)) = the identity transformation +