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

For a locally small category $\mathcal{C}$, the Yoneda lemma states thatnaturally in both $A\in \mathcal{C}$ and $F:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$. Taking $F=\mathcal{C}(-,B)$ givesso the Yoneda embedding is full and faithful.

For small $\mathcal{C}$, the presheaf category has pointwise finite limits and colimits, exponentialand 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.