In quantum field theory (specifically $\phi^4$ theory), $W$ is the sum of all connected Feynman diagrams and the effective action $\Gamma$ is the sum of all 1PI Feynman diagrams. They are related by a Legendre transform. Then $W^{(n)}$ and $\Gamma^{(n)}$ represent $n$-point connected correlation functions and $n$-point 1PI functions obtained from $W$ and $\Gamma$ by taking functional derivatives.
How can one interpret these correlation functions and represent them using diagrams? Is $W^{(n)}$ the sum of all connected diagrams with $n$ external legs? Likewise, is $\Gamma^{(n)}$ the sum of all 1PI diagrams with $n$ external legs?
There also exists an object $\Sigma$, which is the sum of all 1PI diagrams with 2 external legs (Peskin & Schroeder page 219). Does this mean that $\Sigma = \Gamma^{(2)}$?
I often see $\Sigma$ written with an argument, for example $\Sigma(p)$ or $\Sigma(p^2)$, is $p$ the momentum associated with the two external legs?
There already exists a similar question (What is, diagrammatically, the 2-vertex $\Gamma^{(2)}$?) but I do not fully understand the accepted answer, specifically why the diagram they show represents $\Gamma^{(n)}$.