In general, in order to specify a category completely I must specify what are the objects, what are the maps, which object is the domain of each map, which object is the codomain of each map, which map is the identity of each object, and which map is the composite of any two ‘composable’ maps – six things to be specified. Of course, this cannot be done in any arbitrary way. Recall these laws that must be satisfied:
Here is a special case. Suppose we have only one object, which we call ‘*’. This means that all the maps in the category are endomaps (of this unique object). Nevertheless there may be many maps in this category. ...