I'm following along with https://bartoszmilewski.com/2014/12/05/categories-great-and-small/ and trying to make sure I understand the content -- I'm still working out the differences between category theory, and (I think it's called) the category of sets.
Consider arrows as relations (not functions) for the following:
1) a category with one object, and one arrow
2) a category with one object, and multiple arrows
3) a category with multiple objects, and no more than one arrow between objects
1) is a monoid (by definition) and a thin category (by definition). The single object in the category would be a set with no more than one element, and it would have to form a valid relationship back to itself. (Could it have zero elements? I don't know)
2) is a monoid (by definition) but not a thin category (by definition, because the homset has more than one element). The single object in the category would have elements that all have relations with every other element. An example might be a set with a,b,c where a,b,c are equal, and composition with arrows is the relation "less than or equal to."
3) not a monoid because there is more than one object, but it is a thin category because every homset has at most one arrow. You could have composition define a relation like "less than or equal to" and this will form a directed acyclic graph, etc. (a partial order I think)
It seems that outside the case of an object with one element, there isn't a monoid that is also a thin category (in the category of sets, where composition of arrows is a relation).
Is the above statement and my understanding of 1,2, and 3 correct?