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# Exercise 57 - Chapter 3

edited June 2018

Let $$\mathcal{C}$$ be an arbitrary category and let $$\mathcal{P}$$ be a preorder, thought of as a category. Consider the following statements:

1. For any two functors $$F, G : \mathcal{C} \rightarrow \mathcal{P}$$, there is at most one natural transformation $$F \rightarrow G$$.
2. For any two functors $$F, G : \mathcal{P} \rightarrow \mathcal{C}$$, there is at most one natural transformation $$F \rightarrow G$$.

For each, if it is true, say why; if it is false, give a counterexample.

1) Any $$\alpha_c$$ in $$\mathcal{P}$$ is unique, because $$\mathcal{P}$$ is a preorder meaning at most 1 arrow exists between a pair of objects.
2) There is nothing in $$\mathcal{C}$$ restricting the number of arrows between a pair of objects.
Comment Source:1) Any \$$\alpha_c\$$ in \$$\mathcal{P} \$$ is unique, because \$$\mathcal{P} \$$ is a preorder meaning at most 1 arrow exists between a pair of objects. ![One Natural Transformation](https://docs.google.com/drawings/d/e/2PACX-1vSrQVA_5UJFCXyLjDq4aNZehvlM4rWuM7o8UxKPAzl4YihE1daYEkvbws_5G9089Dej9B4N5Z7JcoMB/pub?w=607&h=383) 2) There is nothing in \$$\mathcal{C} \$$ restricting the number of arrows between a pair of objects. ![Two Natural Transformations](https://docs.google.com/drawings/d/e/2PACX-1vR_x8gK_X_cK3UcOTO8PAiMeu2bILBjwg8Fq6fz5g9PaWxkr2XY55VRM1pkAtEsd-k2XWXTmOZOvfrw/pub?w=786&h=406)