Let $C$ be locally small. Consider the Yoneda embedding $Y:C\rightarrow [C^{op},Set]$. Since limits in functor categories are computed pointwise and since the hom-functor preserves limits, the Yoneda embedding is limit-preserving. A natural question to ask then is, whether or when the Yoneda embedding has a left adjoint. Categories whose Yoneda embedding has a left adjoint are called total. Apparently the category of sets and the category of groups are both total.
What are the left adjoints of the Yoneda embedding (of these or other interesting examples) explicitely?
Best Answer
Given a contravariant functor $F:\mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$, we need to find a set $Y$ with a universal natural transformation $F \to \mathrm{Hom}(-,Y)$.
Note that the contravariant functor $\mathrm{Hom}(-,Y)$ (treated as a functor $\mathbf{Set} \to \mathbf{Set}^{\mathrm{op}}$) is left adjoint to itself (treated as a functor $\mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$).
Let $Y=F(1)$ where $1$ is a singleton set. Then, for any set $X$, there is a natural map $F(X) \to \mathrm{Hom}(X,F(1))$ given by the adjunct of the map $X \cong \mathrm{Hom}(1,X) \to \mathrm{Hom}(F(X),F(1))$ under the above adjunction.
To see that this is universal, let $\eta:F \to \mathrm{Hom}(-,Z)$ be a natural transformation. Then, the induced map $F(1) \to Z$ is just $\eta_1$ (modulo the isomorphism $\mathrm{Hom}(1,Z) \cong Z$).
This shows that $\mathbf{Set}$ is total.