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'\$$).