# Why are the morphisms in comma categories not defined the other way around?

A comma category of functors $$S:\mathcal A\to \mathcal C$$ and $$T:\mathcal B\to \mathcal C$$ is a category where objects are pairs $$(A,B,h)$$ with $$h:S(A)\to T(B)$$, and morphisms are pairs $$(f,g):(A,B,h)\to(A’,B’,h’)$$, where the following diagram commutes:

Why isn’t this defined so that $$T(g)$$ is a morphism from $$T(B’)\to T(B)$$ instead? This intuitively seems more reasonable to me, since it would mean that we can interpret morphisms in the comma category as “turning $$h$$ into $$h’$$”. I am just now learning about comma categories so I don’t really have an intuition for them or for why they are used.

• You may find the twisted arrow category interesting. May 22, 2019 at 10:53
• You don't want to "turn $h$ into $h'$", you want $h$ and $h'$ to be "compatible". Mostly, these comma categories generalize the usual comme categories $C/c$, $c/C$, and the comma categories used to define (co)limits, and so the definition follows from this desire to generalize May 22, 2019 at 12:43

One particular example that may interest you is that of natural transformations (nLab, Wikipedia). Given functors $$F, G: \mathcal{C} \to \mathcal{D}$$, a natural transformation $$\eta: F \Rightarrow G$$ is essentially a morphism of functors. It is defined as follows: $$\eta$$ is a collection of arrows in $$\mathcal{D}$$ of the form $$\eta_C: F(C) \to G(C)$$ for each object $$C$$ in $$\mathcal{C}$$. These arrows should satisfy some naturality condition, namely that for every arrow $$f: C \to C'$$ in $$\mathcal{C}$$ we have that $$\require{AMScd} \begin{CD} F(C) @> F(f) >> F(C')\\ @V \eta_C VV @VV \eta_{C'} V \\ G(C) @>> G(f) > G(C') \end{CD}$$ commutes. So this is saying that $$\eta$$ 'is compatible with arrows in $$\mathcal{C}$$'.
You can probably already see the similarity with the diagram you provided. We can actually make that precise, we can reformulate the definition of a natural transformation in terms of a comma category. A natural transformation $$\eta: F \Rightarrow G$$ then corresponds to a functor $$T: \mathcal{C} \to (F \downarrow G)$$ such that $$T(C) = (C, C, \eta_C)$$ and $$T(f) = (f, f)$$. Here $$(F \downarrow G)$$ denotes the comma category. It might be nice to write this out for yourself to see why this is true.
• It's probably worth noting that $(Id_{\mathcal C}\downarrow Id_{\mathcal C})$ corresponds to the arrow category $\mathcal C^2$ where $2$ is the category with two objects and one arrow between them. $\mathcal C^2$ is a pretty important and natural category. General comma categories can then be computed via a limit in $\mathbf{Cat}$ involving $\mathcal C^2$ which realizes a special 2-limit. May 22, 2019 at 18:03