A (small) category consists of a set of objects and a set of arrows
together with the following structure:
- a source map assigning an object to each arrow ,
- a target map: assigning an object t(f) to each arrow ,
- an identity map assigning to each object an arrow with
- a composition map assigning to each pair of arrows , such that , a third arrow with and .
- The composition thus defined "" is associative, that is, whenever these compositions make sense.
- the identity map satisfies for any such that and , and any such that .