Notation in Category Theory, related to total categories, colimit$_{x\in C, z \in P(x)} x$ A definition says:

Call a category C total if the Yoneda embedding has a left adjoint
  $F:$ PShv(C) $ \to $ C.

If $P \in$ PShv(C), then $F(P) \cong$ colimit$_{x\in C, z \in P(x)} x$.
Moreover, a category C is total if this colimit exists.
What is this symbol colimit$_{x\in C, z \in P(x)} x$? Why is there this isomorphism, and why is C total if this colimit exists?
 A: The colimit is really indexed by the category of elements $\int_{\mathcal{C}} P$, whose objects are pairs $(x,z)$ with $x \in \mathrm{ob}(\mathcal{C})$ and $z \in P(x)$, and where a morphism $f : (x,z) \to (x', z')$ is a morphism $f : x \to x'$ in $\mathcal{C}$ such that $P(f)(z') = z$.
Every presheaf $P$ is a colimit of representables indexed by $\int_{\mathcal{C}} P$, namely
$$P \cong \mathrm{colim}_{(x,z) \in \int P} ~ \mathsf{y}(x)$$
in $\mathrm{Psh}(\mathcal{C})$, where $\mathsf{y} : \mathcal{C} \to \mathrm{Psh}(\mathcal{C})$ is the Yoneda embedding.
Since $F$ is a left adjoint, it preserves colimits, meaning that
$$F(P) \cong \mathrm{colim}_{(x,z) \in \int P} ~ F\mathsf{y}(x)$$
Since the Yoneda embedding is full and faithful, the counit $\varepsilon : F \circ \mathsf{y} \to \mathrm{id}_{\mathcal{C}}$ of the adjunction $F \dashv \mathsf{y}$ is a natural isomorphism, and so we obtain
$$F(P) \cong \mathrm{colim}_{(x,z) \in \int P} ~ F\mathsf{y}(x) \cong \mathrm{colim}_{(x,z) \in \int P} ~ x$$
You now need to verify that if this colimit exists for all presheaves $P$, then the assignment $P \mapsto \mathrm{colim}_{(x,z) \in \int P} ~ x$ determines a functor $\mathrm{Psh}(\mathcal{C}) \to \mathcal{C}$ which is left adjoint to the Yoneda embedding.
