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For $b:B\to B^{\prime }$ in $\mathcal{B}$, conjugate the induced natural transformation by the chosen representations of a functor:The Yoneda lemma gives a unique morphism $Gb:GB\to G{B}^{\prime }$ whose postcomposition map is ${\tau }_b$. Explicitly,Consequently, for every $f:A\to GB$,This is precisely the required naturality in $B$. The identities ${\tau }_{1_B}=1$ and ${\tau }_{b^{\prime }b}={\tau }_{b^{\prime }}{\tau }_b$ force $G{1}_B=1_{GB}$ and $G(b^{\prime }b)=G{b}^{\prime }Gb$, because the Yoneda embedding is fully faithful. Thus the representing objects extend uniquely to a functor $G:\mathcal{B}\to \mathcal{A}$. Uniqueness follows by the same Yoneda lemma: any candidate satisfying that naturality must induce ${\tau }_b$, so must have exactly the displayed action on morphisms.

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.

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 giveThe bijections are natural in $X$, soThus the essential image of $\mathcal{C}$ in its presheaf topos is full and closed under finite products and exponentials.

Suppose a small category has equalizers. If a categorical presheaf $P$ is a retract $P\stackrel{i}{\to }{H}_B\stackrel{p}{\to }P$, with $pi=1_P$, then $i$ equalizes $ip$ and $1_{H_B}$. The Yoneda lemma writes $ip=H_b$ for an endomorphism $b:B\to B$. An equalizer $A\to B$ of $b$ and $1_B$ gives the same presheaf equalizer, since the Yoneda embedding preserves categorical limits. Uniqueness of equalizers gives $P\cong H_A$.

For $f:A\to B$, the Yoneda embedding sends $f$ to postcomposition $\mathcal{C}(-,A)\to \mathcal{C}(-,B)$. If this is epic in the presheaf category, its component at $B$ is surjective, so $1_B=fg$ for some $g:B\to A$. Conversely such a section makes the natural transformation split epic. Pointwise surjectivity follows directly by doubling the target presheaf along the image subpresheaf: the two inclusions agree on the image but differ on any element outside it.

For a small Cartesian closed category, the Yoneda embedding into its presheaf category preserves exponential objects. At $C$, the exponential in a presheaf category has value $\mathrm{Nat}(YC\times YA,YB)$. Since $YC\times YA\cong Y(C\times A)$, the Yoneda lemma identifies this with $\mathcal{C}(C\times A,B)\cong \mathcal{C}(C,B^A)$. Every identification is natural in all three objects.