>**Puzzle 121.** Let \$$\mathbf{N}\$$ be the free category on this graph:

>

>It has one morphism for each natural number. What are all the functors \$$F: \mathbf{N} \to \mathbf{N}\$$? What happens when we compose them?

Such a functor, \$$F\$$, must map 0 (\$$id_\mathbf{N}\$$) to 0,

\$F(id_\mathbf{N}) = id_\mathbf{N} = 0, \$

and it must send the non-identity \$$s\$$ to either \$$id_\mathbf{N}\$$, \$$s\$$, or \$$s\$$ repeated a certain finite number of times (\$$s\circ s\circ \cdots \circ s \$$),

\$F(s) =id_\mathbf{N} \$

\$\Leftrightarrow \\\\ F(s\circ s) = F(s)\circ F(s) = id_\mathbf{N} \circ id_\mathbf{N} = 0 + 0, \$

\$\Leftrightarrow \\\\ F(\underbrace{s\circ s\circ \cdots \circ s}\_{n \text{ times }}) \\\\ = \underbrace{F(s)\circ F(s)\circ \cdots \circ F(s)}\_{n \text{ times }} \\\\ =\underbrace{id_\mathbf{N} \circ id_\mathbf{N} \circ \cdots \circ id_\mathbf{N} }\_{n \text{ times }} \\\\ =\underbrace{0 + 0 + \cdots 0}\_{n \text{ times }} = 0 \$

and,
\$F(s) =s = 1 \$
\$\Leftrightarrow \\\\ F(s\circ s) = F(s)\circ F(s) =s \circ s =1+1 , \$
\$\Leftrightarrow \\\\ F(\underbrace{s\circ s\circ \cdots \circ s}\_{n \text{ times }}) \\\\ = \underbrace{F(s)\circ F(s)\circ \cdots \circ F(s)}\_{n \text{ times }} \\\\ =\underbrace{s\circ s\circ \cdots \circ s}\_{n \text{ times }} \\\\ =\underbrace{1 + 1 + \cdots 1}\_{n \text{ times }} = n \$

which makes such an \$$F\$$ the identity functor and,

\$F(s) = \underbrace{s\circ s\circ \cdots \circ s}\_{m \text{ times }} = m \$
\$\Leftrightarrow \\\\ F(s\circ s) = F(s)\circ F(s) = \underbrace{s\circ s\circ \cdots \circ s}\_{m \text{ times }} \circ \underbrace{s\circ s\circ \cdots \circ s}\_{m \text{ times }} = m+m = 2m, \$
\$\Leftrightarrow \\\\ F(\underbrace{s\circ s\circ \cdots \circ s}\_{n \text{ times }}) \\\\ = \underbrace{F(s)\circ F(s)\circ \cdots \circ F(s)}\_{n \text{ times }} \\\\ =\underbrace{ \underbrace{s\circ s\circ \cdots \circ s}\_{m \text{ times }}\circ \underbrace{s\circ s\circ \cdots \circ s}\_{m \text{ times }}\circ \cdots \circ \underbrace{s\circ s\circ \cdots \circ s}\_{m \text{ times }}}\_{n \text{ times }} \\\\ =\underbrace{m + m + \cdots m}\_{n \text{ times }} = m*n \$

which would imply that functors \$$F: \mathbf{N} \to \mathbf{N}\$$ are multiplication.