Left adjoint to a a functor

Question

Let $\delta:Set\to Cat$ be a functor that sends a set to a discrete category whose objects are elements of the set. I see that the right adjoint is the object functor. However, I am struggling to define a left adjoint. I want some hints on how to do this.

Answer 1

The left adjoint to $\delta$ is the "connected components" functor.

Given a category $\mathcal{C}$, we can define an equivalence relation on $\text{ob}(\mathcal{C})$ by taking the symmetric and transitive closure of the relation $xRy$ $\Leftrightarrow$ there is an arrow $X\to Y$. An equivalence class for this relation is called a connected component of $\mathcal{C}$.

Now if $X$ is a set, then a functor $F\colon \mathcal{C}\to \delta(X)$ is completely determined by its action on objects (since $\delta(X)$ has at most one arrow between any two objects), and if $xRy$ in $\mathcal{C}$, then $F(x) = F(y)$ (since there are no arrows in $\delta(X)$ between distinct objects). It follows that $F$ sends every object in a connected component to the same object of $\delta(X)$, so the functors $\mathcal{C}\to \delta(X)$ are in natural bijection with the functions from the connected components of $\mathcal{C}$ to $X$.