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Given morphisms $f:A\to B$ and $g:B\to C$, their composite is $gf:A\to C$. Composition is associative and has the identity morphisms as units.

The objects are the entities between which a category specifies morphisms. Each object has an identity morphism.

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 $L$.