+</pre>
+
+That's it. Well, there may be a wrinkle here. I don't know whether the definition of a monoid requires the operation to be defined for every pair in its set. In the present case, <code>γ <=< φ</code> isn't fully defined on `T`, but only when <code>φ</code> is a transformation to some `MF'` and <code>γ</code> is a transformation from `F'`. But wherever `<=<` is defined, the monoid laws must hold:
+
+<pre>
+ (i) γ <=< φ is also in T
+
+ (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
+
+ (iii.1) unit <=< φ = φ
+ (here φ has to be a natural transformation to M(1C))
+
+ (iii.2) ρ = ρ <=< unit
+ (here ρ has to be a natural transformation from 1C)
+</pre>
+
+If <code>φ</code> is a natural transformation from `F` to `M(1C)` and <code>γ</code> is <code>(φ G')</code>, that is, a natural transformation from `FG'` to `MG'`, then we can extend (iii.1) as follows:
+
+<pre>
+ γ = (φ G')
+ = ((unit <=< φ) G')
+ = (((join 1C) -v- (M unit) -v- φ) G')
+ = (((join 1C) G') -v- ((M unit) G') -v- (φ G'))
+ = ((join (1C G')) -v- (M (unit G')) -v- γ)
+ = ((join G') -v- (M (unit G')) -v- γ)
+ since (unit G') is a natural transformation to MG',
+ this satisfies the definition for <=<:
+ = (unit G') <=< γ
+</pre>
+
+where as we said <code>γ</code> is a natural transformation from some `FG'` to `MG'`.
+
+Similarly, if <code>ρ</code> is a natural transformation from `1C` to `MR'`, and <code>γ</code> is <code>(ρ G)</code>, that is, a natural transformation from `G` to `MR'G`, then we can extend (iii.2) as follows:
+
+<pre>
+ γ = (ρ G)
+ = ((ρ <=< unit) G)
+ = (((join R') -v- (M ρ) -v- unit) G)
+ = (((join R') G) -v- ((M ρ) G) -v- (unit G))
+ = ((join (R'G)) -v- (M (ρ G)) -v- (unit G))
+ since γ = (ρ G) is a natural transformation to MR'G,
+ this satisfies the definition <=<:
+ = γ <=< (unit G)
+</pre>
+
+where as we said <code>γ</code> is a natural transformation from `G` to some `MR'G`.