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I am a beginner in Category Theory and have encountered two doubts in the subject:

Consider $\mathcal U$ to be a category and $X,Y$ be objects in $\mathcal U$. Then if $f:X\to Y$ is a morphism then a section of $f$ is defined to be a morphism $g:Y\to X$ such that $f\circ g=id_Y$.

Is it trivial that this morphism $g$ always exists?

Let $\mathcal U$ and $\mathcal U'$ be two categories and $Fun(\mathcal U,\mathcal U')$ be the (same variance) category of functors from $\mathcal U$ to $\mathcal U'$. Then the morphisms of $Fun(\mathcal U,\mathcal U')$ are defined as follows: if $\lambda,\mu$ are two (assume covariate) functors from $\mathcal U$ to $\mathcal U'$ then a morphism $t:\lambda\to\mu$ is the collection of morphisms $t_X:\lambda(X)\to \mu(X)$ where $X$ varies over all objects in $\mathcal U$.

This $t_X$ is chosen in a way such that for any morphism $f:X\to Y$, $\mu(f)\circ t_X=t_Y\circ \lambda(f)$.

What is the motivation for this last definition of morphisms on $Fun(\mathcal U,\mathcal U')$? An example would be really helpful.

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    $\begingroup$ Here's an example I like : If $n \ge 1$, you can define a functor $\textrm{GL}_n$ from the category of commutative unitary rings to the category of groups (for any ring $A$, $\textrm{GL}_n(A)$ is the groups of $n \times n$ invertible matrices with coefficients in $A$. If $f : A \to B$ is a ring morphism, then $\textrm{GL}_n(f)$ is the morphism obtained by applying $f$ component by component). There is a natural transformation $\det : \textrm{GL}_n \to \textrm{GL}_1$ (where $\det_A : \textrm{GL}_n(A) \to \textrm{GL}_1(A)$ is simply the usual determinant map as you know it). $\endgroup$ – Joel Cohen Aug 22 '16 at 15:08
  • $\begingroup$ This is a really nice example. Thank you for sharing it! $\endgroup$ – Landon Carter Aug 22 '16 at 15:28
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  1. No, sections are not guaranteed to exist, this is just a definition. From another perspective, any morphism admitting a section is an epimorphism, but not all morphisms are epic.

  2. Morphisms between functors are also called natural transformations, and are ubiquitous in mathematics. This notion captures the idea that some construction of an object $G(X)$ from an object $F(X)$ can be performed in a canonical, coordinate-free, simple and clean way. You will find many examples in the wikipedia article. In fact, Saunders Mac Lane once said

I didn't invent categories to study functors; I invented them to study natural transformations

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