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I am getting closer to the essence of a comma category.

Given 1) category $\mathcal{A}$ with objects $A, B$ and signle not-identity morphism $f: A \mapsto B$, and 2) single-object category with single (identity) morphism $\mathcal{1}$, I am trying to construct $(F / G$), where $F$ is an identity functor on the $\mathcal{A}$ and $G: \mathcal{1} \mapsto \mathcal{A}$ always selects, say, the $A \in Obj(\mathcal{A})$.

So there exist two objects: $(A, \circ \in Obj(\mathcal{1}), F(A) = A \mapsto G(\circ) = A)$ and $(B, \circ \in Obj(\mathcal{1}), F(B) = B \mapsto G(\circ) = A)$. Here it feels wrong to me: $F(B) \mapsto G(\circ) = B \mapsto A$ must be some morphism in the $\mathcal{A}$ according to the defintion; however such morphism does not exist.

So, I got two simple questions here: 1) What did I do wrong above? 2) What does comma category say to us? How do we "read" it, what kind of information it packs about participating functors and underlaying categories?

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Given functors $F:\mathcal{C}\to\mathcal{D}$ and $A:1\to\mathcal{D}$, the objects of $(F\downarrow A)$ are pairs $(X,f)$ with $X\in\mathrm{obj}(\mathcal{C})$ and $f\in\mathcal{D}(F(X),A)$. If there aren't any morphisms $F(X)\to A$, then there is no such pair, and thus no such object.

  • In the above example, the only object in $(id_\mathcal{A}\downarrow A)$ is $(A,id_A)$; it essentially the trivial category.

  • In fact if you took the natural numbers and their ordering as a category, and let $0:1\to\mathbb{N}$ be the functor that picks out $0$, then $(id_\mathbb{N}\downarrow 0)$ would also have only one object and the identity arrow; infinitely many objects of $\mathbb{N}$ would play no role whatsoever in the comma category.

  • If we considered $\mathbb{N}$ as a discrete category (where the only morphisms are identities) and let $o:\mathrm{Odd}(\mathbb{N})\to\mathbb{N}$ be the inclusion of the odd numbers, then the comma category $(o\downarrow 0)$ has no objects at all--it is the empty category.

Point being that $\mathcal{C}$ can have all kinds of objects, without there being any guarantee that the comma category is non-empty; much less that every object of $\mathcal{C}$ has some associated object the comma category.

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