The relationships between coordinate systems are represented in AST by Objects called Mappings. A Mapping does not represent a coordinate system itself, but merely the process by which you move from one coordinate system to another related one.
A convenient picture of a Mapping is as a ``black box'' (see Figure below) into which you can feed sets of coordinates.
For each set you feed in, the Mapping returns a corresponding set of transformed coordinates. Since each set of coordinates represents a point in a coordinate space, the Mapping acts to inter-relate corresponding positions in the two spaces, although what these spaces represent is unspecified. Notice that a Mapping need not have the same number of input and output coordinates. That is, the two coordinate spaces which it inter-relates need not have the same number of dimensions.
In many cases, the transformation can, in principle, be performed in either direction: either from the input coordinate space to the output, or vice versa. The first of these is termed the forward transformation and the other the inverse transformation.
Further reading: For a more complete discussion of Mappings,
AST A Library for Handling World Coordinate Systems in Astronomy