In the beginning, it is mentioned that initially setting all morphisms would determine the full property for each object as well. In this section, we will observe some properties using functors called representables, which consider how each object sees or is seen via morphisms by the fixed object. Furthermore, we learn about the Yoneda lemma, which is nontrivial but a fundamental theorem of category theory. This tells us a lot of information about the structure of the presheaf category.

To make the whole logic clear, every category in this section is assumed to be locally small.

Representables

We begin with the definition of the functor $H^A$, where $A$ is an object from $\mathcal{A}$.

$$ \begin{gather*}

H^A:\mathcal{A}\rightarrow\textbf{Set}:B\mapsto\mathcal{A}(A,B),\,g\mapsto g\circ - \end{gather*} $$

Here, if $g$ is a morphism from $B$ to $C$ in category $\mathcal{A}$, then $H^A(g)$ is a well-defined function from $\mathcal{A}(A,\,B)$ to $\mathcal{A}(A,\,C)$, and thus $H^A$ is a well-defined functor.

Simply speaking, since the functor sends each object to the set of morphisms from $A$ to the object, this shows the world seen by $A$.

Dually, we can define the functor $H_B$, dealing with the world being encoded in $B$.

$$ \begin{gather*} H_B:\mathcal{A}^\text{op}\rightarrow\textbf{Set}:A\mapsto\mathcal{A}(A,B),\,f\mapsto-\circ f \end{gather*} $$

These functors and the functors naturally isomorphic to them are called representables, and the choice of corresponding object $A$ and natural isomorphism $\alpha$ together are called the representation of the representable.

Here are some basic examples of representables:

• In section 3-1, it says that $1$ represents elements, and $2$ represents subsets. Its precise meaning is that functors $1_\textbf{Set}$ and $\mathcal{P}$ are naturally isomorphic to $H^1$ and $H_2$, respectively. Generalizing the categorical definition of the element to an arbitrary category, we say a morphism from the fixed object $S$ to the object $A$ is a generalized element of $A$ of shape $S$. For example, loops in topological space are generalized elements of shape $S^1$.
• Adjunctions also induce representables. Since $\mathcal{A}(A,\,G(B))\cong\mathcal{B}(F(A),\,B)$ naturally in either $A$ and $B$, it induces natural isomorphisms between $H^A\circ G$ and $H^{F(A)}$, $H_B\circ F$ and $H_{G(B)}$. Thus, $H^A\circ G$ and $H_B\circ F$ are representables. If $\mathcal{A}=\textbf{Set}$, we plug $A=1$ to get $H^A\circ G\cong 1_\textbf{Set}\circ G=G$, therefore $G$ itself is also representable.
• In linear algebra, the functor $\textbf{Billin}(U,V;-)$, which sends objects to the set of bilinear maps from $U\times V$ to the object, is naturally isomorphic to $H^{U\otimes V}$, whence representable.

By the definition, $H_A$ is a presheaf on $\mathcal{A}$, and $H^A$ is a presheaf on $\mathcal{A}^\text{op}$. Therefore, we can think of $H_$ and $H^$ as sending each object to the corresponding object in the proper functor category. Indeed, these are well-defined functors.

$$ \begin{gather*} H^:\mathcal{A}^\text{op}\rightarrow [\mathcal{A},\textbf{Set}]:A\mapsto H^A=\mathcal{A}(A,-),\,f\mapsto H^f=-\circ f\\ H_:\mathcal{A}\rightarrow[\mathcal{A}^\text{op},\textbf{Set}]:B\mapsto H_B=\mathcal{A}(-,B),\,g\mapsto H_g=g\circ- \end{gather*} $$

Although those definitions may seem complicated, they are essentially identical with the homset-assinging functor $\text{Hom}_\mathcal{A}:\mathcal{A}^\text{op}\times\mathcal{A}\rightarrow\textbf{Set}$ described below, just like the way the function $f:A\times B\rightarrow C$ is interpreted as $f:A\rightarrow(B\rightarrow C)$ or $f:B\rightarrow(A\rightarrow C)$.

$$ \text{Hom}_\mathcal{A}:\mathcal{A}^\text{op}\times\mathcal{A}\rightarrow\textbf{Set}:(A,\,B)\mapsto\mathcal{A}(A,\,B)(f,\,g)\mapsto g\circ-\circ f $$