By slightly generalizing the definition from Awodey's Category Theory, I obtained:
(this is almost exactly what is on Awodey 75-76 with a few slight modifications). And the first diagram is the same as in Awodey's definition of group objects but without the "i" "inversion" arrow. I had just subconsciously assumed earlier that this "1" is a zero object" but then I realized that I was told that $\textbf{Grp}$ = Group($\textbf{Set}$), and $\textbf{Set}$ has no zero object iirc (I think $\emptyset$ is initial and the singleton is terminal). So maybe it is supposed to be a generic terminal object? I am suspecting it should either be terminal or initial (but not both for the reason I just mentioned), but not really sure which one it should be (I think the zero group [monoid] is a zero object in $\textbf{Grp}$ [$\textbf{Mon}$] [respectively], which is why I am not really sure if we want it to be initial or terminal in general). Maybe we don't require it to be initial or terminal?