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.