>**Puzzle 107.** Take the free category on this graph:


>


>and then impose the equation \\(s \circ s \circ s \circ s = 1_z\\). You get a category with one object, also known as a **[monoid](https://en.wikipedia.org/wiki/Monoid)**. How many morphisms does this category have? How is it related to the picture below?

>

To simplify the proof, let,
\\[
s^{\circ n}=\underbrace{s\circ s\circ s\circ\cdots s}\_{n \text{ times}} \\\\
\\#\lbrace s^{\circ n} \rbrace= \\#\lbrace f \mid f=s^{\circ n}\rbrace
\\]


When \\(n=1\\),
\\[
s^{\circ 1}=s \\\\
\\#\lbrace s^{\circ 1} \rbrace= 1.
\\]
When \\(n=2\\),
\\[
s^{\circ 2}= s\circ s \\\\
\\#\lbrace s^{\circ 2} \rbrace= 1.
\\]

When \\(n=3\\),
\\[
s^{\circ 3}=s\circ s\circ s \\\\
\\#\lbrace s^{\circ 3} \rbrace= 1.
\\]

When \\(n=4\\),
\\[
s^{\circ 4}=s\circ s\circ s\circ s=1_z \\\\
\\#\lbrace s^{\circ 4} \rbrace= 0.
\\]
Summing up all the terms,
\\[
3=\sum_{n=1}^{4}\\#\lbrace s^{\circ n} \rbrace
\\]
Giving an answer of \\(3\\).

Edit: Are we also counting \\(1_z\\)? Otherwise, this gives a different answer.