Construction 8.5.9.3 (Concatenation). Let $\mathrm{Dy}$ denote the Thompson groupoid. We define a functor $\circledast : \mathrm{Dy} \times \mathrm{Dy} \rightarrow \mathrm{Dy}$ as follows:
On objects, the functor $\circledast $ is given by the formula
\[ [0,s] \circledast [0,t] = [0, s+t ]. \]On morphisms, the functor $\circledast $ is given by the formula
\[ (f \circledast g)(x) = \begin{cases} g(x) & \text{ if } 0 \leq x \leq t \\ f(x-t) + s & \text{ if } t \leq x \leq s+t. \end{cases} \]
We will refer to $\circledast : \mathrm{Dy} \times \mathrm{Dy} \rightarrow \mathrm{Dy}$ as the concatenation functor on the Thompson groupoid $\mathrm{Dy}$. Note that the operation $\circledast $ is strictly associative, and admits a (strict) unit given by the degenerate interval $\{ 0\} = [0,0]$. Consequently, $\circledast $ determines a strict monoidal structure on the category $\mathrm{Dy}$ (in the sense of Definition 2.1.2.1).