For a <locally small category> $\mathcal C$, the <Yoneda lemma> states that
$$
\operatorname{Nat}(\mathcal C(-,A),F)\cong F(A),
\qquad \alpha\longmapsto\alpha_A(1_A),
$$
naturally in both $A\in\mathcal C$ and $F:\mathcal C^{\mathrm{op}}\to\mathbf{Set}$. Taking $F=\mathcal C(-,B)$ gives
$$
\operatorname{Nat}(yA,yB)\cong\mathcal C(A,B),
$$
so the <Yoneda embedding> is full and faithful.

For small $\mathcal C$, the <presheaf category> has pointwise finite limits and colimits, exponential
$$
(G^F)(C)=\operatorname{Nat}(yC\times F,G),
$$
and a <subobject classifier> whose elements at $C$ are sieves on $C$. Hence it is a <presheaf topos>.

If $\mathcal C$ has finite limits, the Yoneda embedding preserves them: maps into a limiting object are the corresponding limits of hom-sets. Its essential image is therefore a full subcategory of the presheaf topos closed under finite limits. Since Yoneda is full and faithful, this proves the assertion up to equivalence.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 119 / 6 / a / Solution

Suppose the small category $\mathcal C$ is <Cartesian closed>. The <Yoneda embedding> preserves finite products. For $A,B,X\in\mathcal C$, the <Yoneda lemma> and the <exponential object> adjunction give
$$
(yB)^{yA}(X)
=\operatorname{Nat}(yX\times yA,yB)
\cong\mathcal C(X\times A,B)
\cong\mathcal C(X,B^A)
=y(B^A)(X).
$$
The bijections are natural in $X$, so
$$
\boxed{(yB)^{yA}\cong y(B^A).}
$$
Thus the essential image of $\mathcal C$ in its <presheaf topos> is full and closed under finite products and exponentials.

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 119 / 6 / b / Solution

{c}
{title2=$y:\mathcal C\to[\mathcal C^{\mathrm{op}},\mathbf{Set}]$}
{wiki}

The Yoneda embedding sends $A$ to $\mathcal C(-,A)$. The <Yoneda lemma> identifies natural transformations $yA\to yB$ with morphisms $A\to B$, so the embedding is <full and faithful>. It preserves every limit that exists in $\mathcal C$.

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 Incoming links: Yoneda embedding