Via Theorem 3.90:

\\(V=\\{1\\}\\). \\(\mathrm{lim}\_\textbf{1}D=\\{(d_1)|d\_1\in D(1)\,\mathrm{and}\,D(\mathrm{id}\_1)(d_1)=d_1\\}\\) together with a projection map \\(p_1((d_1))=d_1\\). However, since \\(D\\) is a functor, the constraint is always satisfied, and the projection map is the identity. So, the limit of \\(D:\textbf{1}\to\textbf{Set}\\) is the set that it picks out in **Set**.

Via Definition 3.87:

Let \\(S\in\textbf{Set}\\) be the set that \\(D\\) picks out. A cone over \\(D\\) consists of a set \\(C\\) and a function \\(f:C\to S\\) (the cone property is trivial here, just like the constraint in theorem 3.90). Morphisms between these cones are functions \\(a:C\to C'\\) such that for \\(g:C'\to S\\), \\(g\circ a=f\\). A terminal object in this category **Cone**(\\(D\\)) is a set \\(T\\) and a function \\(h:T\to S\\) such that for every set \\(C\\) and function \\(f:C\to S\\) there is a unique \\(b:C\to T\\) with \\(h\circ b=f\\). For each element \\(s\in S\\) there is a cone \\(\\{ * \\},f\_s:\\{ * \\}\to S\\}\\) where \\(f_s(*)=s\\). Each \\(b_s\\) such that \\(h\circ b_s=f_s\\) must pick out an element in \\(T\\), since the images of each \\(f_s\\) are distinct, the images of each \\(b_s\\) must also be distinct, so \\(\lvert T\rvert\geq\lvert S\rvert\\) and \\(h\\) must be surjective. Uniqueness of \\(b_s\\) forces \\(h\\) to be injective, so \\(h\\) is an isomorphism and \\(T\cong S\\). This shows that following defintion 3.87 agrees with following theorem 3.90.

\\(V=\\{1\\}\\). \\(\mathrm{lim}\_\textbf{1}D=\\{(d_1)|d\_1\in D(1)\,\mathrm{and}\,D(\mathrm{id}\_1)(d_1)=d_1\\}\\) together with a projection map \\(p_1((d_1))=d_1\\). However, since \\(D\\) is a functor, the constraint is always satisfied, and the projection map is the identity. So, the limit of \\(D:\textbf{1}\to\textbf{Set}\\) is the set that it picks out in **Set**.

Via Definition 3.87:

Let \\(S\in\textbf{Set}\\) be the set that \\(D\\) picks out. A cone over \\(D\\) consists of a set \\(C\\) and a function \\(f:C\to S\\) (the cone property is trivial here, just like the constraint in theorem 3.90). Morphisms between these cones are functions \\(a:C\to C'\\) such that for \\(g:C'\to S\\), \\(g\circ a=f\\). A terminal object in this category **Cone**(\\(D\\)) is a set \\(T\\) and a function \\(h:T\to S\\) such that for every set \\(C\\) and function \\(f:C\to S\\) there is a unique \\(b:C\to T\\) with \\(h\circ b=f\\). For each element \\(s\in S\\) there is a cone \\(\\{ * \\},f\_s:\\{ * \\}\to S\\}\\) where \\(f_s(*)=s\\). Each \\(b_s\\) such that \\(h\circ b_s=f_s\\) must pick out an element in \\(T\\), since the images of each \\(f_s\\) are distinct, the images of each \\(b_s\\) must also be distinct, so \\(\lvert T\rvert\geq\lvert S\rvert\\) and \\(h\\) must be surjective. Uniqueness of \\(b_s\\) forces \\(h\\) to be injective, so \\(h\\) is an isomorphism and \\(T\cong S\\). This shows that following defintion 3.87 agrees with following theorem 3.90.