Recall that a small category is a category where the class of objects is a set. As a result, the class of morphisms is also a set. One can thus define a small category completely within set theory, as a 6-tuple $(O,M,s,t,i,c)$ , where

1. $O$ is the set of objects and $M$ is the set of morphisms
2. $s,t: M\to O$ are functions such that $s(f)$ is the source (domain) of $f$ , and $t(f)$ is the target (codomain) of $f$
3. $i:O\to M$ is a function such that $i(A)$ is the identity morphism $1_A$
4. $c:K\to M$ is a function such that $c(g,f)$ is the composition of morphism $f$ followed by morphism $g$ (or $g\circ f$ ); here, $K$ is the collection the all composable pairs of morphisms: $$K=\lbrace (g,f)\in M\times M\mid s(g)=t(f)\rbrace$$

1. the source and target of an identity morphism on an object $A\in O$ is just $A$ : $$s(i(A))=t(i(A))=A$$
2. the source of $c(g,f)$ is the the source of $f$ , and the target of $c(g,f)$ is the target of $g$ : $$s(c(g,f))=s(f)\qquad \mbox{ and }\qquad t(c(g,f))=t(g)$$
3. the composition of a morphism $f$ with the identity morphism of its source $s(f)$ is just $f$ ; same holds for $t(f)$ : $$c(f,i(s(f)))=f=c(i(t(f)),f)$$
4. composition is associative, if defined: that is, if $(g,f),(h,g)\in K$ , then $$c(h,c(g,f))=c(c(h,g),f)$$

An internal category is the ``categorical abstraction'' (and generalization) of a small category. Whereas a small category can be completely described in Set, the category of sets, an internal category is completely specified within another category, using only objects and morphisms of this category and their properties.

Definition. Given a category $\mathcal{C}$ with pullbacks, an internal category (or category object) $\mathcal{D}$ of $\mathcal{C}$ consists of the following:

1. two objects $O,M$ of $\mathcal{C}$ , where $O$ is called the object of objects, and $M$ the object of morphisms,
2. two morphisms $s,t:M\to O$ , where $s,t$ are called the source and target respectively,
3. a morphism $i:O\to M$ called the identity,
4. a morphism $c:M\times_O M\to M$ called the composition, where $M\times_O M$ is the pullback of $s$ and $t$ :
   

1. $s\circ i=t\circ i=1_O$ , the identity morphism on $O$
2. $s\circ c = s\circ p_2$ and $t\circ c=t\circ p_1$

3.

$c\circ \displaystyle{i\!\circ\! t \choose 1_M} = 1_M = c\circ \displaystyle{i_M \choose i\!\circ\! s}$

Since $M \times_O (M \times_O M) \cong (M\times_O M)\times_O M \cong M\times_O M \times_O M$ , we may view $M\times_O M \times_O M$ as the domain of morphisms $t \times_O 1_M$ and $1_M \times_O s$ . We are now ready for condition 4:

4.

$c\circ (t \times_O 1_M) = c\circ (1_M \times_O s)$ .

Remark. In Set, an internal category is just a small category as we have seen from the discussion earlier. An internal category in Cat is a double category.