For questions related to Yoneda lemma in category theory, which basically says (among many other things it says) that every locally small category can be embedded nicely into it's functor category into the category of Sets. Use this tag along with (abstract-algebra) or (category-theory).
For a locally small category $\mathcal{C}$ and any two objects $A$ and $B$ of $\mathcal{C}$, the collection of all morphisms from $A$ to $B$, denoted $\mathrm{Hom}(A,B)$ is a set. With this in mind, for each object $A$ we get a functor $h^A \colon \mathcal{C} \to \text{Set}$ where each object $X$ of $\mathcal{C}$ gets sent to the set $\mathrm{Hom}(A,X)$. This functor $h^A$ is often more tersely denoted as $\mathrm{Hom}(A,{-})$. In summary, each object $A$ gives you a functor $h^A$.
Now let $\text{Set}^\mathcal{C}$ denote the category of all functors from $\mathcal{C}$ to $\text{Set}$; for two functor $F$ and $G$, the morphisms from $F$ to $G$ are the natural transformations from $F$ to $G$, and will be denoted $\mathrm{Nat}(F,G)$. Using the previous idea, we can define a single functor $よ \colon \mathcal{C} \to \text{Set}^\mathcal{C}$ that sends an object $A$ to its functor $h^A$.
Yoneda Lemma — For any object $A$ of $\mathcal{C}$ and any functor $F$ in $\text{Set}^\mathcal{C}$, there is a natural isomorphism of sets $\mathrm{Nat}(h^A,F) \cong F(A)$.
Corollary — The functor $よ$ is fully faithful. For this reason $よ$ is called the Yoneda embedding.
You could tell an analogous story using the contravariant functor $h_A = \text{Hom}({-},A)$ instead.
