The identity morphism \\(Id_A: A \rightarrow A\\) for object A is defined by the requirement that it is a 'no-op' with respect to composition.
This can be expressed as follows. Suppose \\(T\\) is a morphism tree, which contains \\(Id_A: A \rightarrow A\\) as a subtree. Let \\(T'\\) be the result of 'splicing' \\(Id_A\\) out of the tree.
Then composite(\\(T\\)) = composite(\\(T'\\)).