I want to draw diagrams for each of fundamental notions in basic category theory: categories, functors and natural transformations. While I have been able to do this for categories and functors, I cannot do the same for natural transformations. My diagrams for categories and functors are as follows ($\mathcal{C}_1$ is the class of morphisms, $\mathcal{C}_0$ the class of objects of a category $\mathcal{C}$)

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Is there a way to make a picture of a natural transformation $\alpha$ between two parallel functors $F,G:\mathcal{C}\rightrightarrows\mathcal{D}$ in the same style of the previous diagrams for categories and functors?

  • Note: I am not looking for the naturality square of a natural tranformation, which I already know. I am looking for a (tridimensional?) diagram involving $\alpha, F,G,\mathcal{C},\mathcal{D}$.

A natural transformation $\alpha$ from $F: \mathcal{C} \to \mathcal{D}$ to $G : \mathcal{C}\to \mathcal{D}$ consists of a map $\alpha : C_0 \to D_1$ such that $d_D \alpha = F_0$ and $c_D \alpha = G_0$ satisfying $\circ_D \langle G_1 , \alpha d_{C} \rangle = \circ_ D \langle \alpha c_C, F_1\rangle$ where $d_C$ and $c_C$ are what you called $\text{dom}_{C}(\ )$ and $\text{cod}_C(\ )$ respectively, and $\langle G_1,\alpha d_C\rangle : C_1 \to D_2$ and $ \langle \alpha c_C, F_1\rangle : C_1 \to D_2$ are the expected maps into the pullback (i.e. $\langle G_1,\alpha d_C\rangle(f) = (G(f),\alpha(\text{dom}_{C}(f))$). I hope this is understandable without diagrams.

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  • $\begingroup$ the only part I can't understand is when you talk about the pullback. Which pullback is it? $\endgroup$ – Danae Kissinger Jul 29 '16 at 20:03
  • $\begingroup$ $D_2$ can be constructed as the pullback of $\text{cod}_D(\ )$ and $\text{dom}_{D}(\ )$. $\endgroup$ – Nex Jul 29 '16 at 20:44
  • $\begingroup$ ok now it's clear $\endgroup$ – Danae Kissinger Jul 29 '16 at 20:49

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