I am aware that forgetful functors usually have a left adjoint, but I'm more interested in developing my technique.
Let $\mbox{Cat}$ be the category of small categories and $\mbox{Set}$ the category of sets.
Show that the forgetful functor $U:\mbox{Cat}\to \mbox{Set}$ has a left adjoint.
We need a functor $F:\mbox{Set} \to\mbox{Cat}$ such that we have an adjunction with $F$ as left and $U$ as right adjoint.
An adjunction is determined by finding an assignment $F_0 (X) \in \mbox{Cat}_0, X\in\mbox{Set}_0,$ and for every $X\in \mbox{Set}_0$ a universal morphism $$\eta _A : A\to U(F_0(X)) $$ from object $X$ to functor $U$. More precisely, this universal morphism is an (the) initial object of the category $(X\downarrow U)$, where $$(X\downarrow U) _0 = \left\lbrace (h,\mathcal C) \mid \mathcal C \in \mbox{Cat}_0 \ \&\ h : X \to \mathcal C_0 \right\rbrace $$ and a morphism $(h,\mathcal C) \to (k,\mathcal D)$ is a functor $f :\mathcal C\to \mathcal D$ such that $$ (U(f)h)(x) = k(x),\quad x\in X. $$
For a fixed set $X$ the initial object is surely something that involves the identity map on $X$. But there's a problem. We must have a (small) category whose set of objects is $X$. Denote this as $\mathcal C^X$. The universal morphism would be $\mbox{id}_X : X\to \mathcal C_0^X = X$.
- Question. How do we pick category $\mathcal C^X$?
Once we have that sorted we can define $F :\mbox{Set} \to \mbox{Cat}$ s.t $F(X) = F_0(X)$ and for a map $v : X\to Y$ the morphism (functor) $F(v) : \mathcal C_X\to\mathcal C_Y$ would have to satisfy $$U(F(v)) \eta _X = \eta _Y v \quad \mbox{i.e in this case}\quad (UF)(v) = v $$
While I am aware of something like constructing a vector space from a (non-empty) set (freeness), I don't know what this would mean for small categories. How to make progress?
The left adjoint to $U$ is the "discrete category functor" $F=D$, which sends a set $X$ to the discrete category $D(X)=D_X$ with the set $X$ as the set of objects, and a mapping $f\colon X\to Y$ to the functor $D(f)=D_f\colon D_X\to D_Y$ between discrete categories, such that $f$ is the mapping on objects of $D_f$. It is obvious that $D\colon\mathbf{Set}\to\mathbf{Cat}$ is indeed a functor. Now the unit mapping of adjunction $(D,U)$ is simply an identity mapping $id(X)\colon X\to U(D_X)$.