Given a category $C$ with limits and colimits. Let $F: D \to C$ be a connected diagram in $C$. I want to show that if we just take the iterated pushouts then I will eventually get the colimit. For example if my diagram was $x \leftarrow u \rightarrow y \leftarrow v \rightarrow z$, then by taking the pushout $x \cup_u y$ and $y \cup_v z$. And then take the pushout $(x \cup_u y) \cup_y (y \cup_v z)$ this gives me the colimit of the diagram. With a small diagram like this it is easy to show but how would I go about showing this is true in general. I know by the Yoneda embedding and the Kan extension, I can just deal with presheafs. We can view each object in the diagram as a presheaf by the Yoneda embedding $Y : D \to \text{Set}^{D^{op}}$.
So now I just need to deal with a diagram of presheafs. The colimit of a diagram is just a weighted colimit when the weight is 1 on every object in the diagram. I want to show that the colimit of these presheafs is exactly the the presheaf that sends every object to 1. Sorry if I'm not clear anywhere. I'm quite new to Category theory.