Are all diagrams commutative?

Question

I begin to learn category theory. The following three definitions are provided.

Definition 1. Two morphisms $f\colon a \to b$ and $g\colon c \to d$ are the same if and only if $a = c$ and $c = d$.

Definition 2. A diagram is a collection of objects and morphisms from a category.

Definition 3. A diagram is called commutative, if any composable sequence of morphisms with the same endpoints (of a sequence) results in the same composite.

Then, as the endpoints of sequences are the same, the sources and the targets of the composites are the same, so are all diagrams commutative? I suspect at least one definition is not correct, or I don't understand some parts of them.

Answer 1

The first definition doesn't make sense. Given two objects in a category, there can be many distinct morphisms from one to the other. To answer your main question: no, not every diagram is commutative. For example, in the category of sets, a diagram could look like this:

In this example, you have three sets $X,Y,Z$, as well as functions $f:X\to Y$, $g:Y \to Z$, and $h:X \to Z$. Commutativity of the diagram means that $h = g\circ f$. This is certainly not the case if you take specific functions $f,g$ and an arbitrary $h$ (write down explicit examples).

Answer 2

The definition of a category does not say whether two morphisms are equal in general. This is only given for specific composites containing identity. Therefore, the first definition is not general, thus not every diagram is commutative.

You can define a general category without specifying what other mathematical objects the objects and morphisms are. Then, you cannot always determine whether a diagram is commutative, because you have no way of determining whether two composites are the same, that is, checking the third definition. Perhaps you can say that if you cannot check the definition, then it is not fulfilled, thus a diagram is not commutative.

An example of a category in which you can tell whether any diagram is commutative without specifying what other mathematical objects the objects and morphisms are, is any category that contains only identity morphisms. There, any diagram commutes.

An example of a specific category where a diagram does not commute because of the notion of equality is given by @AlexProvost in his answer.

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This answer is how I would like the question to be answered, but it is based on the effort of others. I would like to thank you all for comments and answers, but especially @AlexProvost, @Joe, and @NoahSchweber.