Some doubts in Category Theory

Question

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.

Answer 1

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