I have a commutative diagram: $f_*\circ \phi = \psi \circ Df$. I have seen commutative diagrams with a circle in the middle. But what does the direction of the arrows $\circlearrowleft$ vs $\circlearrowright$, and how do I determine which is suitable for this diagram?
EDIT2: It is possible that the direction of the error does not mean anything.
EDIT1: Here is an example of the sort of thing I am referring to:
Typically, diagrams drawn with an arrow (oriented in any direction) indicate commutativity of the given diagram - the orientation of the arrow itself is irrelevant. You might also see a circle in the center of a diagram used to indicate the same thing. However, I think it is more common to simply state in words that the diagram is commutative.
In fact, this notation has become so prevalent that if I see any diagram drawn where there are multiple paths from one point to another, I almost automatically assume the diagram is commutative! Personally, I get the willies whenever I draw a diagram that doesn't commute, and I think if you encounter a non-commutative diagram in the wild, the author will (or at least should) make it abundantly clear that it doesn't commute.