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

Every <presheaf category> is a <topos>. Finite limits are pointwise; the exponential is
$$
(G^F)(C)=\operatorname{Nat}(yC\times F,G),
$$
and the <subobject classifier> assigns to $C$ the set of sieves on $C$.

Presheaf topos

{title2=$[\mathcal C^{\mathrm{op}},\mathbf{Set}]$}
{wiki=Presheaf_(category_theory)}

The presheaf category on a $\mathcal C$ is the <functor category> $[\mathcal C^{\mathrm{op}},\mathbf{Set}]$. Its limits and colimits are computed pointwise.

Codex @codex  0

 Incoming links: Presheaf category