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A natural transformation between presheaves assigns maps $F(U)\to G(U)$ commuting with every restriction. For presheaves of abelian groups or modules, those maps preserve the specified algebraic structure.

A natural bijection between set-valued functors is a natural transformation whose component functions are bijections. Their inverse functions are automatically natural, so it is a natural isomorphism.

A natural transformation all of whose components are isomorphisms. Their inverses form the inverse natural transformation.

For a natural transformation $\theta :F\Rightarrow G$, naturality is the equation $G(f){\theta }_A={\theta }_{B}F(f)$ for each morphism $f:A\to B$. Thus transforming and then applying a functor gives the same result as applying that functor and then transforming. It makes the component maps a coherent transformation of functors.

A natural transformation between functors $F,G:\mathcal{C}\to \mathcal{D}$ assigns a categorical morphism ${\alpha }_A:F(A)\to G(A)$ to every object of a category $A$, withThis naturality condition says that transforming the input before transporting it gives the same result as transporting it first and transforming the output. It constrains the whole family of arrows, so unrelated choices of a map for each object generally fail it.

For a concrete example, compare the functor that leaves every set and function unchanged with the power set functor using images of subsets. The maps ${\eta }_X:X\to \mathcal{P}(X)$ given by $x\mapsto \{x\}$ form a natural transformation: for any function $f:X\to Y$,The right-hand panel of the category illustration checks the square at $x=3$ with $f(x)=x+1$: both routes end at $\{4\}$. The statement works for every function, rather than depending on a preferred labeling of a particular set.

Let $F:\mathcal{C}\to \mathcal{D}$, $G:\mathcal{D}\to \mathcal{C}$, $\eta :1\Rightarrow GF$ and $\epsilon :FG\Rightarrow 1$ be natural transformations. If $G\epsilon {\eta }_G=1_G$, then $e={\epsilon }_{F}F\eta :F\Rightarrow F$ is an idempotent morphism in the functor category. Naturality implies $Ge\eta =\eta$ and $\epsilon e_G=\epsilon$; these absorption identities prove $e^2=e$. The omitted triangle measures exactly whether $e=1_F$.

A natural transformation $\tau :F\Rightarrow G$ satisfying $G_2{\tau }_{X\otimes Y}=({\tau }_X\otimes {\tau }_Y)F_2$ and $G_0{\tau }_I=F_0$.

The two formulas for the adjunction transposes give both triangle identities for an adjunction directly. Apply the inverse bijection to the transpose of $1_{FA}$:Apply the forward bijection to the inverse transpose of $1_{GB}$:These are equalities of components of natural transformations, soNo cancellation assumption on either functor is needed: both identities follow from the inverse hom-set bijections.

Write the adjunction as the natural bijectionThe unit of an adjunction and counit of an adjunction are the natural transformations whose components areNaturality of the hom-set bijections gives, for $f:FA\to B$ and $g:A\to GB$,For example, postcomposition by $f$ in the first hom-set corresponds to postcomposition by $Gf$ in the second, giving the first formula; naturality in $A$ gives the second. The same naturality says that, for $a:A\to A^{\prime }$ and $b:B\to B^{\prime }$,Thus these components do define the asserted natural transformations $1_{\mathcal{C}}\to GF$ and $FG\to 1_{\mathcal{D}}$.

Let $\hat{\mathcal{O}}=[\mathcal{O}(S{)}^{\mathrm{op}},\mathbf{Set}]$, with categorical presheaf restrictions $X(U)\to X(V)$ when $V\subseteq U$. The five functors areIn $\Delta A$ all restriction maps are identities. In $\Lambda A$ and $\nabla A$, all nonidentity restrictions are the unique maps permitted by the displayed sets. These definitions are valid also when $A$ is empty: a restriction out of $(\nabla A)(S)$ still has singleton target. Maps of sets act at the displayed $A$-components, and natural transformations act on $\Pi$ and $\Gamma$ by evaluation.

A natural transformation $\Lambda A\to X$ is determined by an arbitrary map $A\to X(\varnothing )$, since all other source components are empty. A natural transformation $X\to \Delta A$ is determined by its empty-open component $X(\varnothing )\to A$: the component at $U$ must first restrict $X(U)\to X(\varnothing )$. Similarly, a natural transformation $\Delta A\to X$ is determined by $A\to X(S)$, followed by the restriction $X(S)\to X(U)$. Finally, a natural transformation $X\to \nabla A$ is determined by $X(S)\to A$, since all proper-open targets are singletons. These descriptions give the hom-set bijectionsThus the five adjoints for constant presheaves on open sets areThe values at the empty open set are genuine presheaf values here; no sheaf condition or sheafification is being imposed.

Here a universal element means a pair $(A,u)$, with $u\in X(A)$, such that for every $B$ and every $x\in X(B)$ there is a unique morphism $f:B\to A$ satisfying $X(f)(u)=x$. Equivalently,By the Yoneda lemma these component maps constitute a natural transformation $h_A\to X$. If $u$ is universal, every component is bijective, so this is a natural isomorphism and $X$ is a representable presheaf. Conversely a natural isomorphism $h_A\to X$ takes $1_A$ to an element $u$ with exactly this property. In the category of elements of the contravariant functor $X$, $(A,u)$ is a terminal object: every $(B,x)$ has a unique arrow to it.

Let $\mathcal{C}$ be a locally small category, let $h_A=\mathcal{C}(-,A)$ be a representable presheaf, and let $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$. The Yoneda lemma gives a bijection, natural in $A$ and $X$,For $x\in X(A)$, define a natural transformation ${\alpha }^x:h_A\to X$ byFor $v:C\to B$, the functor composition law gives $X(v){\alpha }_B^x(f)=X(fv)(x)={\alpha }_C^x(fv)$, proving naturality. Conversely, naturality of any $\alpha :h_A\to X$ along $f:B\to A$ givesThus $\alpha$ is determined by its displayed element, and ${\alpha }_A^x(1_A)=x$ proves that the constructions are inverse.

A natural transformation $\theta :X\to Y$ sends the element to ${\theta }_A(x)$, which corresponds to $\theta {\alpha }^x$. A morphism $u:A^{\prime }\to A$ induces $h_u:h_{A^{\prime }}\to h_A$; precomposition by $h_u$ sends the element to $X(u)(x)$. This proves naturality in both variables and completes the Yoneda lemma proof.

Let $a:X\to Y$ be a natural transformation of categorical presheaves. If every $a_C:X(C)\to Y(C)$ is a surjective function, and $r,s:Y\rightrightarrows Z$ satisfy $ra=sa$, then $r_C{a}_C=s_C{a}_C$ at every $C$. Surjectivity gives $r_C=s_C$ for all $C$, hence $r=s$. Thus $a$ is an epimorphism.

Conversely suppose $y\in Y(C)$ is outside the image of some $a_C$. Construct the pushout witness for failure of pointwise surjectivity explicitly: at each $D$ takeRestrictions send $[(z,k)]$ to $[(Y(f)z,k)]$; naturality of $a$ ensures that every generating identification is respected. Thus $P$ is a categorical presheaf, and the two inclusions $j_0,j_1:Y\to P$ are natural transformations with $j_{0}a=j_{1}a$.

All identifications concern the same element $z$ in its two copies, so an element outside the image is never identified with its other copy. Consequently $j_{0,C}(y)\ne j_{1,C}(y)$, proving $a$ is not an epimorphism. ThereforeThis is the pointwise epimorphism in a functor category criterion. It requires neither a componentwise choice of preimages nor a natural section.