Sentence:  The <WSD>axioms</WSD> read as follows. For every composable pair "f" and "g" the composite f \circ g goes from the domain of "g" to the codomain of "f". For each object "A" the identity arrow 1_A goes from "A" to "A". Composing any arrow with an identity arrow (supposing that the two are composable) gives the original arrow. And composition is associative. 
Meaning of the word marked with <WSD> tags:  A fundamental assumption that serves as a basis for deduction of theorems; a postulate (sometimes distinguished from postulates as being universally applicable, whereas postulates are particular to a certain science or context).