Is there a difference between a pushout and a copullback?

Question

I've started reading up on category theory more, and I'm confused about whether a colimit also inverts the category where the functor starts. To illustrate, which of these would be a pushout, and which a copullback?

Answer 1

If I understand your diagrams correctly, the bottom right one is a pushout.

First, 'copullback' is not a very standard name. So if you don't define what you mean by that, it is difficult to answer the question of whether this is the same as a pushout.

Next, to elaborate on the relation between a limit and a colimit: the colimit of a diagram $D:\mathsf J \to \mathcal C$ is isomorphic to the limit of the diagram $D^{\rm op}:\mathsf J^{\rm op} \to \mathcal C^{op}$ defined as follows

1. an object $j$ in $\mathsf J^{\rm op}$ is the same as an object of $\mathsf J$, and $D^{\rm op}$ maps it to $D(j)$ (this is an object of $\mathcal C$, hence also of $\mathcal C^{\rm op}$)
2. a morphism $j\to k$ in $\mathsf J^{\rm op}$ is the same as a morphism $f:k \to j$ in $\mathsf J$ and $D^{\rm op}$ maps it to $D(f)$ (this is an arrow $D(k) \to D(j)$ in $\mathcal C$, hence also a morphism $D^{\rm op}(j) \to D^{\rm op}(k)$ in $\mathcal C^{\rm op}$)