# Questions tagged [natural-transformations]

Questions concerning morphisms of functors or unnatural isomorphisms.

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### Among morphisms of morphisms, what makes commutative squares special?

Given two (1-)categories $\mathcal{C}, \mathcal{D}$, and given the 0-category (class) of funtors $\mathcal{C} \to \mathcal{D}$, denoted $Func(\mathcal{C} \to \mathcal{D})$, let's say we want to make ...
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### Proof and explanation of a vector space not being naturally isomorphic to its dual

From https://ncatlab.org/nlab/files/SelingerSelfDual.pdf It is well-known that each finite dimensional vector space A is isomorphic, but not naturally isomorphic, to its dual space A∗. I would like ...
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### What are colored morphisms/arrows intended to mean in these diagrams?

I've been reading more category theory as a prerequisite to understanding some more complicated theorems, and for the first time I'm running into arrows that are distinctly colored. Examples include ...
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### Natural isomorphism between the complexification of $V$ and $V^2$, when $V$ is already a complex vector space?

I'm considering the question of: What happens if you complexify a vector space that is already complex? I basically believe the following. I take complexification of a complex vector space to mean ...
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### Understanding the solution of Exercise 4.1.32 in Tom Leinster "Basic Category Theory".

Here is the exercise and its solution: 1-Is there a typo and $\varphi$ should be $\psi$? 2- I do not understand how by exercise 2.1.14 we will get the first equation in the solution of 4.1.32. And ...
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### Natural Isomorphism $(Y^Z)^X\cong Y^{Z\times X}$ in a cartesian closed category

Let $\mathcal{C}$ be a cartesian closed category. I'm working on a problem that asks me to show that for $X,Y,Z\in\text{ob}(\mathcal{C})$ there is a natural isomorphism $(Y^Z)^X\cong Y^{Z\times X}$. ...
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