Abbreviation: Cat
A \emph{category} is a structure $\mathbf{C}=\langle C,\circ,\text{dom},\text{cod}\rangle$ of type $\langle 2,1,1\rangle$ such that $C$ is a class,
$\langle C,\circ\rangle$ is a (large) partial semigroup
dom amd cod are total unary operations on $C$ such that
$\text{dom}(x)$ is a left unit: $\text{dom}(x)\circ x=x$
$\text{cod}(x)$ is a right unit: $x\circ\text{cod}(x)=x$
if $x\circ y$ exists then $\text{dom}(x\circ y)=\text{dom}(x)$ and $\text{cod}(x\circ y)=\text{cod}(y)$
$x\circ y$ exists iff $\text{cod}(x)=\text{dom}(y)$
Remark: The members of $C$ are called \emph{morphisms}, $\circ$ is the partial operation of \emph{composition}, dom is the \emph{domain} and cod is the \emph{codomain} of a morphism.
The set of objects of $C$ is the set $\mathbf{Obj}C=\{\text{dom}(x)|x\in C\}$. For $a,b\in C$ the set of homomorphism from $a$ to $b$ is $\text{Hom}(a,b)=\{c\in C|\text{dom}(c)=a\text{ and }\text{cod}(c)=b\}$.
Let $\mathbf{C}$ and $\mathbf{D}$ be categories. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a homomorphism: $h(\text{dom}(c))=\text{dom}h(c)$, $h(\text{cod}(c))=\text{cod}h(c)$ and $h(c\circ d)=h(c) \circ h(d)$ whenever $c\circ d$ is defined.
Morphisms between categories are called \emph{functors}.
Example 1: The category of function on sets with composition.
In fact, most of the classes of mathematical structures in this database are categories.
$\text{dom}(\text{dom}(x))=\text{dom}(x)=\text{cod}(\text{dom}(x))$
$\text{cod}(\text{cod}(x))=\text{cod}(x)=\text{dom}(\text{cod}(x))$
$\begin{array}{lr}
f(1)= &1\\ f(2)= &3\\ f(3)= &11\\ f(4)= &55\\ f(5)= &329\\ f(6)= &2858\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\
\end{array}$