Abbreviation: Grpd

A \emph{groupoid} is a category $\mathbf{C}=\langle C,\circ,\text{dom},\text{cod}\rangle$ such that

every morphism is an isomorphism: $\forall x\exists y\ x\circ y=\text{dom}(x)\text{ and }y\circ x=\text{cod}(x)$

Let $\mathbf{C}$ and $\mathbf{D}$ be Schroeder categories. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a \emph{functor}: $h(x\circ y)=h(x)\circ h(y)$, $h(\text{dom}(x))=\text{dom}(h(x))$ and $h(\text{cod}(x))=\text{cod}(h(x))$.

Remark: These categories are also called \emph{Brandt groupoids}.

Example 1:

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

$\begin{array}{lr}

f(1)= &1\\
f(2)= &2\\
f(3)= &3\\
f(4)= &7\\
f(5)= &9\\
f(6)= &16\\
f(7)= &22\\
f(8)= &42\\
f(9)= &57\\
f(10)= &90\\

\end{array}$

http://oeis.org/A140189

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