'In category theory, a diagram commutes means that between any two objects in the diagram, there is a unique morphism. And the identity morphisms and compositions of morphisms might be omitted in the diagram.'
I can understand the first half of this paragraph. However, the meaning of the last half is vague. My question is very explicit, as follows:
If there is one object, which appears in the diagram twice, is there an implicit identity morphism between its twice appearance? And does it mean that all the compositions of morphisms from object $X$ to object $X$ is the identity morphism of $X$?
Best Answer
In general I would assume that a diagram said to commute and containing the same object twice does not have an implicit identity between those appearances of the same object.
You do get the same object twice in examples such as those shown in: https://en.wikipedia.org/wiki/Monad_(category_theory) with the diagrams defining a monad. In one diagram $T^2$ appears twice with no identity being intended. In the other both $T^2$ and $T$ appear twice with an explicit identity on $T$ but definitely no intention that there is one on $T^2$.