hom-set definition of limit?

Question

I've heard of limits of a diagram as a cone with universal property. But on the ncatlab website, they define it as a hom-set. Specifically, for the limit of a set valued functor $F: D^{op}\rightarrow Set,$ we have $LimF:=Hom_{[D^{op},Set]}(pt, F)$ where $pt:D^{op}\rightarrow Set$ by $d \mapsto \{*\}.$ They call $pt$ the functor constant on a point. So is $pt$ the constant functor? What is $D^{op}?$ Why is it an opposite category? Usually we have a functor $F:I\rightarrow C,$ where $I$ is the indexing category and can be thought of as a diagram, while $F$ assigns objects to the dots and morphisms in the arrows of the diagram. I don't see how this definition is equivalent to the definition of limit using cones.

Answer 1

Denote $\hat {\mathcal D}$ the category $[\mathcal D^{\mathrm {op}},\mathsf{Set}]$ (we call it the category of presheaves on $\mathcal D$). Denote also $\mathfrak h \colon \mathcal D \to \hat{\mathcal D}$ the Yoneda embedding $d \mapsto \hom_{\mathcal D}(-,d)$. Then, for any presheaf $F \colon \mathcal D^{\mathrm{op}} \to \mathsf{Set}$, Yoneda's lemma states that there is an isomorphism $$ F \simeq \hom_{\hat{\mathcal D}}(\mathfrak h(-), F) $$ in the category $\hat{\mathcal D}$. So, in $\mathsf{Set}$, $$ \begin{aligned} \lim F &\simeq \lim \hom_{\hat{\mathcal D}}(\mathfrak h(-), F) \\ &\simeq \hom_{\hat{\mathcal D}}(\operatorname{colim}\mathfrak h(-), F). \end{aligned}$$ You can go ahead and verify that $\operatorname{colim}\mathfrak h(-)$ is actually the constant presheaf (this is again Yoneda's lemma).