> **Puzzle 106**. What are some reasonable equations between morphisms that we might want to impose?

Maybe we want to limit the number of layers of management to say 4.
$\text{Manager} \circ \text{Manager} \circ \text{Manager} \circ \text{Manager} = 1_\text{Employee}$

**Puzzle 106-FPE** We want to ensure management has a strict hierarchy, i.e. a tree.
What equations will enforce that?

> **Puzzle 107** How many morphisms does this category have? How is it related to the square?

The morphisms are \$$\lbrace [1_z = s \circ s \circ s \circ s], [s], [s \circ s], [s \circ s \circ s] \rbrace \$$ which are 4.

If \$$s\$$ represents a \$$90 \deg\$$ spin of the square then we have a morphism from the initial pose to each symmetric pose of the square [without flipping/turning], of which there are 4.

$\begin{array}{c|c} \text{Pose} & \text{morphism} \\\\ \hline \tt{0 \deg} & 1_z = s \circ s \circ s \circ s \\\\ \tt{90 \deg} & s \\\\ \tt{180 \deg} & s \circ s \\\\ \tt{270 \deg} & s \circ s \circ s \end{array}$