Let $X$ be path-connected and let $x\in X.$ Let $\mathscr{O}=\{U_\alpha\}_{\alpha \in I}$ be a cover of $X$ by path-connected open sets, each containing $x$. Further suppose $\mathscr{O}$ is closed under finite intersection.
Page 18 of J. P. May's $\textit{A Concise Course in Algebraic Topology}$ states the Van Kampen Theorem as follows:
"View $\mathscr{O}$ as a category with the inclusion maps as morphisms. Then $\pi_1(X, x)$ is the colimit of the diagram $\pi_1(-, x):\mathscr{O} \rightarrow \textbf{Grp}.$"
Here is the standard statement of the Van Kampen Theorem we all know and love:
For $\alpha, \beta \in I,$ let $i_{\alpha\beta}:\pi_1(U_\alpha\cap U_\beta, x)\rightarrow \pi_1(U_\beta, x)$ be the homomorphism induced by the inclusion $U_\alpha\cap U_\beta \rightarrow U_\beta,$ and let $j_\alpha:\pi_1(U_\alpha, x) \rightarrow \pi_1(X, x)$ be the homomorphism induced by the inclusion $U_\alpha \rightarrow X.$ Define the homomorphism $\Phi: \text{FreeProduct}_\alpha(\pi_1(U_\alpha, x)) \rightarrow \pi_1(X, x)$ by mapping a word $\omega_{\alpha_1}...\omega_{\alpha_n}$ (where $\omega_{\alpha_i}\in U_{\alpha_i}$) to the concatenation of loops $(j_{\alpha_1}\omega_{\alpha_1})...(j_{\alpha_n}\omega_{\alpha_n}).$ Then $\Phi$ is surjective and the kernel of $\Phi$ is the normal subgroup $N$ generated by words of the form $i_{\alpha\beta}(\omega)i_{\beta\alpha}(\omega)^{-1}.$ Hence, $\text{FreeProduct}_\alpha(\pi_1(U_\alpha, x))/N \cong \pi_1(X, x).$
I am new to category theory, so I am not seeing the full picture of how these two statements relate to each other. I am looking for any insight on this.
$\textbf{Edit:}$ So far, I have found that if $\mathscr{O}=\{U_\alpha, U_\beta, U_\alpha\cap U_\beta\},$ then the colimit of the diagram $\pi_1(-, x): \mathscr{O} \rightarrow \textbf{Grp}$ is the pushout of $i_{\alpha\beta}$ and $i_{\beta\alpha}$ which turns out to be the free product with amalgamation. This basically answers my question, but I am still looking for good sources that pushouts in $\textbf{Grp}$ are free products with amalgamation.