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For a locally small category $\mathcal{C}$, an object $C$, and a categorical presheaf $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$, the Yoneda lemma gives the natural bijectionIts two maps are explicitlyFor $u:A^{\prime }\to A$, the equation $X(u)X(f)(x)=X(fu)(x)$ proves naturality of $\Psi (x)$. Evaluating it at the identity morphism gives $\Phi \Psi (x)=x$. Conversely, naturality of $\alpha$ at $f:A\to C$ givesso $\Psi \Phi (\alpha )=\alpha$. Evaluation at the identity and transport of an element along a morphism are mutually inverse.

The bijection is natural in both variables: a natural transformation $\theta :X\Rightarrow Y$ sends $\Phi (\alpha )$ to ${\theta }_C(\Phi (\alpha ))$, matching $\Phi (\theta \alpha )$; and $g:C\to C^{\prime }$ givesFor completeness, the covariant Yoneda lemma for $Y:\mathcal{C}\to \mathbf{Set}$ is $\mathrm{Nat}(\mathcal{C}(C,-),Y)\cong Y(C)$, with $\alpha \mapsto {\alpha }_C(1_C)$ and inverse $y\mapsto (f:C\to A\mapsto Y(f)(y))$.

For the categorical presheaf $X$, its category of elements has objects $(C,x)$ with $x\in X(C)$. A morphism $(C,x)\to (D,y)$ is a morphism $f:C\to D$ satisfying $X(f)(y)=x$. Composition in a category is inherited from $\mathcal{C}$: if also $X(g)(z)=y$, then $X(gf)(z)=X(f)X(g)(z)=x$. The identity morphisms are inherited as well. The forgetful functor sends $(C,x)$ to $C$ and $f$ to $f$.

A universal element is a pair $(R,r)$ for which each $x\in X(C)$ is uniquely of the form $X(f)(r)$ for $f:C\to R$. Thus $(R,r)$ is a terminal object of the category of elements, with the variance appropriate to a categorical presheaf.

Given a universal element, defineThe defining uniqueness makes each map a bijection; $X(u){\psi }_C(f)={\psi }_{C^{\prime }}(fu)$ gives naturality for $u:C^{\prime }\to C$. Hence $\psi$ is a natural isomorphism and $X$ is a representable presheaf. Conversely, from a natural isomorphism $\psi :\mathcal{C}(-,R)\cong X$, take $r={\psi }_R(1_R)$. The Yoneda lemma gives ${\psi }_C(f)=X(f)(r)$; its bijectivity makes $(R,r)$ a universal element. Therefore the two descriptions coincide:

Fix the chosen categorical limit object $LA$ and its categorical cone $p_{A,j}:LA\to (DA{)}_j$. For $a:A\to A^{\prime }$, the familyis a categorical cone, by naturality of $Da$. The universal property of $L{A}^{\prime }$ supplies a unique morphismThe identity morphism of $LA$ satisfies the equations for $a=1_A$, so $L{1}_A=1_{LA}$. For composable $a,a^{\prime }$, the equations for $L{a}^{\prime }La$ agree with those for $L(a^{\prime }a)$; uniqueness gives $L(a^{\prime }a)=L{a}^{\prime }La$. Hence the chosen limits define a functor $L:\mathcal{A}\to \mathcal{C}$. This argument also handles an empty indexing category, when each chosen categorical limit is a terminal object.

Equivalently, $LA$ represents the categorical cone functor $C\mapsto \mathrm{Nat}({\Delta }_{J}C,DA)$, and functoriality of chosen representations supplies the same $L$.

A binary categorical product of objects of a category $A,B$ is an object $P$ with categorical morphisms $p_A:P\to A$ and $p_B:P\to B$. Its universal property is that, for every object $X$ and pair of arrows $f:X\to A$, $g:X\to B$, there is exactly one arrow $h:X\to P$ satisfyingThe arrows $p_A,p_B$ are called the product projections. Existence lets us combine two maps; uniqueness means there is no extra unspecified information in the combined map.

In the Category of sets, take $A=\{0,1\}$ and $B=\{a,b,c\}$. Their Cartesian product contains six ordered pairs, and the projection maps select the first and second coordinates. Let $X=\{u,v,w\}$, $f(u)=0$, $f(v)=1$, $f(w)=1$, and $g(u)=a$, $g(v)=b$, $g(w)=a$. The unique required map is $h(u)=(0,a)$, $h(v)=(1,b)$, $h(w)=(1,a)$. Reading either coordinate recovers the corresponding input map.

In a partially ordered set viewed as a category, the projection arrows mean $P\le A$ and $P\le B$. The universal property says that every common lower bound $X$ satisfies $X\le P$, so $P$ is the greatest lower bound. For subsets ordered by inclusion this is $A\cap B$, the intersection; it is a different construction from the Cartesian product in the Category of sets because the allowed arrows have changed.

The universal property also compares any two proposed products: each supplies a unique map to the other compatible with the projections. The two composites have the same projections as the corresponding identity morphisms, so uniqueness makes them identities. Thus a categorical product is determined up to a unique projection-preserving isomorphism, which explains why we can use whichever concrete construction is convenient.

A section of a morphism $p:E\to B$ in a category is a morphism $s:B\to E$ such that $p\circ s=1_B$, where $1_B$ is the identity morphism. Thus a section is a right inverse of $p$; the equation $s\circ p=1_E$ is not required.

For the projection $p:B\times F\to B$, choosing $f_0\in F$ gives $s(b)=(b,f_0)$. Then $p(s(b))=b$, while $s(p(b,f))=(b,f_0)$ may differ from $(b,f)$. This example also explains sections of a fiber bundle: they choose one point in every fibre compatibly with the structure. A section of a morphism is a monomorphism, since $s\circ u=s\circ v$ implies $u=p\circ s\circ u=p\circ s\circ v=v$.