How does one define a group with commutative diagrams? I am currently working through McLarty's book on Elementary Categories, Elementary Toposes. In Chapter 3, he considers a group as an object in a category with a unit map, a multiplication and an inverse map, defined as 
$$e: 1 \to G, \quad m: G \times G \to G, \quad \_^{-1}: G \to G$$ such that the following diagrams commute:


Here are my questions: 


*

*I don't quite understand the definition of $e$. It maps the terminal object (which would have to be the subgroup of 1 element) into $G$. Why not define $e: G \to G$ since it takes every element to itself? 

*Why isn't the first diagram two-sided to account for $e \cdot x = x = x \cdot e$?

*Why are the two maps of the second diagram the same? Since we want to claim that $x\cdot x^{-1} = e = x^{-1} \cdot x$, shouldn't one of the top maps be $\left< \_^{-1}, G\right>$?

*I'm assuming the map ``$G$" means the unique identity map. Is this correct? 

 A: Yes, there are two typos/inconsistent notations in the diagrams (as you found in your questions 3 and 4).


*

*Well, yes, the terminal object plays the role of the 'one-element set' in this abstraction, and a map $1\to X$ plays the role of 'selecting an element' of $X$. The map $e$ wants to select the identity element. 
You are right that one can also consider the constant homomorphism $G\to G$, but observe that this factors through the terminal object as $G\to 1\overset e\to G$.

*The question is justified, however the other equation follows from the other conditions. Can you translate the classic proof?


$\quad xe=xe=xx^{-1}x=ex=x$


*I guess, the top right arrow is indeed intended as $\langle \_^{-1},\,\mathrm{Id}_G\rangle$. 
However, even one of these follows from the rest (it is enough to require a left unit and left invertibility) :


Let $e$ be the left identity, $y$ be a left inverse of $x$ and $z$ a left inverse for $y$, then $y$ is also right inverse of $x$ as 
  $\quad xy=exy=zy\,xy=zey=zy=e$


*Indeed, the maps denoted as $G$ wants to mean the identity $\mathrm{Id}_G$.

