Definition. Category [dm-000G]

A category Cconsists of a collection of things (objects) and a collection of arrows between these things (morphisms). If there is a morphism A \xrightarrow {f} B and a morphism B \xrightarrow {g} C, we also have the composed morphism A \xrightarrow {g \circ f} C. This composition operator must be associative. Additionally each object X has an identity morphism X \xrightarrow {\mathsf {id}_X} X, such that for an arrow X \xrightarrow {a} Y, a \circ \mathsf {id}_X = \mathsf {id}_Y \circ a = a.