I am trying to understand the pushforward and pullback of maps in category theory by applying it to groups, kernels of homomorphisms, and quotient groups. Here is the definition of a pushforward and pullback that I am referring to:
I attempted to create these diagrams to visualize what is going on with normal groups. Below $\pi$ denotes the projection map, $\varphi$ is a homomorphism, $Ker(\varphi)$ is the normal subgroup with respect to $\varphi$, and $\psi$ and $\psi^{-1}$ is the isomorphism between $Ker(\varphi)$ and $G/Ker(\varphi)$.
Does this even make sense what I am trying to look at, or did I go down a deep rabbithole? This post is also to help check my sanity :(