The presheaf $X \in\widehat{A}$ is the colimit of the functor $\varphi_X := h \circ \pi_X$ ,

\pi_X : \int X \to A \\ \pi_X(a, s) := a

where $h$ is the yoneda embedding, $\int X$ is the category of elements of $X$ . For morphism $\varphi_X(u) := u$ .

Proof. [local-0]

We first show that $X$ is a cocone, for each object $(a, s) \in \int X$ , there is a morphism

h_a \xrightarrow{s} X

here abuse notation that $s \in X_a$ has a corresponding $h_a \to X$ in $\widehat{A}$ because the Yoneda lemma.

and for each $(a,s)\xrightarrow{u}(b,t)$ , the following diagram commutes

hence $X$ is a cocone. Given any other cocone $Y$ , which means a collection of sections

f_s : h_a \to Y

where $u^*(f_t) = f_s$ for each $u : (a,s) \to (b,t)$ by definition. The point is if we define a natural transformation

\eta_a : X_a \to Y_a \\ \eta_a(s) := f_s

naturality follows because $X(u)(t) = s$ implies $Y(u)(f_t) = f_s$ . It is clear that $\eta$ is the unique natural transformation under $\varphi_X$ , showing that $X$ is the colimit.