CatT 1.2.5

1 files changed, 12 insertions, 2 deletions

diff --git a/provider/posts/simmons-intro-to-cat-t/1.2.md b/provider/posts/simmons-intro-to-cat-t/1.2.md
index 3e83b5f..b91cac4 100644
--- a/provider/posts/simmons-intro-to-cat-t/1.2.md
+++ b/provider/posts/simmons-intro-to-cat-t/1.2.md
@@ -106,10 +106,20 @@ Topological spaces $(S, \mathcal{O}_S)$, where $S$ is a Set and $\mathcal{O}_S \
 </div>
 
 <div class="exercise">
-Let $A$ be an object of a category $\ca{C}$.
-Show that $\hom{\ca{C}}{A}{A}$ is a monoid under composition.
+Show that $\End{\ca{C}}{A}$ is a monoid under composition, where $A$ is an object of a category $\ca{C}$.
 
 <div class="proof">
+$\circ$ is associative and total on $\End{\ca{C}}{A}$ by the definition of category and $\End{\ca{C}}{A}$ is obviously closed under $\circ$.
 
+$\idarr{A}$ is required to exist by the definition of category and indeed an identity of $\circ$ on $\End{\ca{C}}{A}$.
+</div>
+
+<div class="corollary">
+Each monoid $(M, \cdot)$ is a category ([again](./1.1.md)).
+
+<div class="proof">
+We construct a category with exactly one object $\ast$ and associate to every element $m \in M$ an arrow $\arr{\ast}{m}{\ast}$.
+We further define $m \circ m = m \ast m$.
+</div>
 </div>
 </div>