I am concerned with a question: could comma category be seen as a weaker version (or just a generalization?) of a natural transformation between two functors?
Assume: $F, G : \mathbb{A} \mapsto \mathbb{B}$ are functors.
The comma category $(F | G)$, among other objects, would contain triples of the kind $(F(A), G(A), f: F(A) \rightarrow G(A))$, where $f$ could be seen as the $\lambda_A$-component of the $\lambda: F \implies G$.
AFAICS, a morphism $(F(A), G(A), \lambda_A) \rightarrow (F(B), G(B), \lambda_B)$ in the $(F | G)$ is a naturality requirement itself, just written quite unsually. Since $\lambda$ requires each $(A \rightarrow B) \in Mor(\mathbb{A})$ to commute, $(F|G)$ yields a valid natural transformation iff $\forall f: A \rightarrow B, f \in Mor(\mathbb{A}): \exists f_{(F|G)}: (F(A), G(A), \lambda_A) \rightarrow (F(B), G(B), \lambda_B), f_{(F|G)} \in Mor((F|G)), f_{(F|G)} = (F(f), G(f))$
…in other cases, there might be no complete natural transformation, but it still feels similar and looks like a kind of weaker, more general version of it. Is it right view?
Best Answer
Comma categories do satisfy a particular universal property in the 2-category $\mathbf{Cat}$. Given functors $F:A\to C$ and $G:B\to C$, the comma category $(F\downarrow G)$ comes equipped with the obvious projections $$p_1:(F\downarrow G)\to A$$ and $$p_2:(F\downarrow G)\to B$$ and a natural transformation $$\alpha:F\circ p_1\Rightarrow G\circ p_2$$ which are universal for such diagrams; i.e. given any other functors $X:D\to A$ and $Y:D\to B$ and a natural transformation $\beta:FX\Rightarrow GY$, there is a unique functor $H:D\to (F\downarrow G)$ with $$p_1\circ H=X$$ $$p_2\circ H=Y$$ and $$\alpha H=\beta.$$
It should be easy to see, and a sense of this is probably what prompted your observation, that the comma category $(id_C\downarrow id_C)$ is really just the arrow category on a category $C$, and the above property specializes to the familiar fact that functors into the arrow category of $C$ correspond to natural transformations between pairs of functors to $C$. If we think of arrow categories as spaces of natural transformations, and comma categories as generalized arrow categories, then you're spot on that comma categories are spaces indexing certain classes of natural transformations (or natural transformation-like things).