Exercise 3.20 in Rotman's Algebraic Topology has always reminded me of the phrasing of the Seifert Van-Kampen Theorem. The exercise is as follows
Let $X$ be a space with basepoint $x_0$, and let $\{U_j:j\in J\}$ be an open cover of $X$ of path connected subspaces such that: (i) $x_0 \in U_j$ for all $j$;
(ii) $U_j \cap U_k$ is path connected for all j,k.
Prove that $\pi_1(X,x_0)$ is generated by the subgroups $\text{im } i_{j*}$, where $i_j:(U_j, x_0) \hookrightarrow (X,x_0)$ is the inclusion.
This reminds me of the Seifert Van-Kampen that also related the first homotopy group of $X$ with the homotopy groups of two open and path-connected subspaces $U_1,U_2$. I can see that the exercise deals with more subspaces, but doesn't deal directly with pushout diagrams or the groups $\pi_1(U_j,x_0)$ themselves, while Seifert Van-Kampen deals with only two subspaces but has a formulation in general categorical language. Beyond that, are there any more connections between the two here? If so what are the ways they are similar and different? Thank you so much!