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## Schroeder categories

Abbreviation: SchrCat

### Definition

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

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

##### Morphisms

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 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 groupoids.

Example 1:

### Properties

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.

Classtype first-order class

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &3\\ f(5)= &7\\ f(6)= &9\\ f(7)= &16\\ f(8)= &22\\ f(9)= &42\\ f(10)= &57\\ \end{array}$