+<pre>
+ ((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]
+</pre>
+
+So our **(lemma 1)** is:
+
+<pre>
+ ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)),
+ where as we said γ is a natural transformation from G to MG'.
+</pre>
+
+
+Next recall that `unit` is a natural transformation from `1C` to `M`. So for elements `C1` in <b>C</b>, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism <code>f:C1→C2</code> in <b>C</b>:
+
+<pre>
+ (4) unit[C2] ∘ f = M(f) ∘ unit[C1]
+</pre>
+
+Next, consider the composite transformation <code>((M γ) -v- (unit G))</code>:
+
+<pre>
+ (5) ((M γ) -v- (unit G)) assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
+</pre>
+
+Next, consider the composite transformation <code>((unit MG') -v- γ)</code>:
+
+<pre>
+ (6) ((unit MG') -v- γ) assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
+</pre>