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The Yoneda lemma and the definition of the nerve of a category giveThus such a map is precisely a diagram of objects and composable morphismsin $\mathcal{C}$. The remaining edges and higher faces record the composites forced by this string.

A natural transformation between functors to the Category of sets is a monomorphism in the functor category exactly when every component is an injective function. Componentwise cancellation proves sufficiency. For necessity, the Yoneda lemma converts two elements with equal component images into two transformations from a covariant representable functor; monicity cancels these transformations, forcing the elements to coincide. The argument remains valid for a locally small category in an ambient universe where the functor category is formed.

For a small category $\mathcal{C}$ and $F:\mathcal{C}\to \mathbf{Set}$, the Yoneda lemma gives a canonical pointwise-surjective natural transformationEach representable is projective because evaluation preserves pointwise epimorphisms, and their coproduct is projective. Thus every set-valued functor on a small category is an epimorphic image of a projective object.

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 $P\cong H_A$.

An object with the displayed property is called small projective; the term 0-presentable here uses this all-small-colimits convention. In a presheaf category, every representable presheaf has this property by the Yoneda lemma and pointwise computation of colimits. Conversely the canonical colimit presentation of a presheaf shows that the identity on such an object factors through one representable, making it a retract. If the indexing category has equalizers, a representable retract is again representable by the representable retract criterion.

For a small category $\mathcal{C}$ with binary coproducts, exponentiation by $h_A=\mathcal{C}(A,-)$ in $[\mathcal{C},\mathbf{Set}]$ is naturally $F\mapsto F(-⨿A)$: the exponential at $B$ is $\mathrm{Nat}(h_B\times h_A,F)\cong \mathrm{Nat}(h_{B⨿A},F)\cong F(B⨿A)$ by the Yoneda lemma. This precomposition functor has a right adjoint given by Right Kan extension, so $h_A$ is a tiny object.

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 $\mathcal{C}$.

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.