We want to characterize the limit and colimit of a functor $D\colon J\to \mathcal C$ by generalized elements. The existence of limits theorem states that the limit of $D$ is the equalizer of $$s,t\colon \prod_{j\in Ob(J)}D(j)\to \prod_{f\in Ar(J)}D(\text{cod}(f))$$ where $s=\langle Df\circ\pi_{\text{dom}(f)} \rangle_{f\in Ar(J)}$ and $t=\langle \pi_{\text{cod}(f)}\rangle_{f\in Ar(J)}$. Equalizers are monic, and so we may think of the limit of $D$ as a subobject of the product $\prod D(j)$.
Moreover, the Yoneda embedding $y\colon \mathcal C\to Set^{\mathcal C^{op}}$ preserves all limits. Therefore, it is tempting to write the limit of $D$ by generalized elements (i.e. $x\in X$ means $x\in Ar(\mathcal C)$ with cod$(x)$=X) as $$ \lim D = \left[x\in\prod D(j)| s(x)=t(x)\right]. $$
Unfortunately, the Yoneda embedding does not preserve colimits. However, the colimit of $D$ is a quotient object of the coproduct and I still feel tempted to write, $$ \text{colim } D = \frac{\coprod D(j)}{\sim}, $$ where $\sim = \bigcap [E:\text{equivalence relation} |\ \langle s(x), t(x)\rangle\in E]$ where the equivalence relations are taken on hom-sets. (And $s, t$ here are the dualized arrows of the maps above).
Is this the wrong characterization? Is there a good analogy between the colimit of an arbitrary functor and the colimit in sets? Is $$ \frac{\mathcal C(\square,\coprod D(j))}{\sim}\colon \mathcal C^{op}\to Set $$ a well-defined functor which is representable, i.e., isomorphic to $\mathcal C(\square,\text{colim } D)$?