A more succinct group object diagram (all axioms in one connected diagram), questions about its properties...

Solution 1:

Group theory is algebraic, which means that it is a Lawvere Theory. This essentially means that the category of groups is precisely given by the functor category $[D,Set]_{\text{f.p.p.}}$ of finite product preserving functors, where $D$ is the diagram you have written. In particular, the functors must preserve the terminal object and products.

Solution 2:

The answer to the first question is no. To see this more clearly, note that asserting that a morphism is of the form, say, $m\times\mathrm{id}$, means 1) that the codomain is equipped with two projection maps, which composed with the given morphism yield $m$ and $\mathrm{id}$, and 2) that the two projection maps form a limiting cone (or rather, are supposed to be sent to a limiting cone; a diagram in which certain cones are supposed to be sent to limiting cones is known as a sketch). Thus, your diagram is strictly speaking incomplete: in full detail, it would include the projection morphisms that define the product structures, and the prescriptions of the corresponding compositions with them.

The answer to the second question is also no. This is because the commutative square looks like a cone (with vertex $G^3$) over a diagram ($G^2\xrightarrow mG\xleftarrow mG^2$). Then a cone over the outer square, considered as a given cone over a diagram, is the same data as a cone over that diagram that factors through the given cone. Consequently, the limiting cone over the outer square is simply the cone with vertex the upper-right corner of the square (the vertex of the square considered a sa cone), with the identity map to itself and the rest of the square's morphisms (morphisms in the diagram) to the other objects. More generally, any cofiltered diagram (any pair of morphism $B\to D\leftarrow C$ in the diagram can be completed to a commutative square by morphisms $B\leftarrow A\to C$, and any parallel pair of morphisms $B\to C$, $B\to C$, can be completed to a fork by a morphism $A\to B$) that is finite looks like a cone over a diagram, hence has a limit given by the vertex of that cone.