Let $\mathscr C$ be the category of groups and $G\in \mathscr C$. Let $$m:G\times G \to G,e:1\to G, i:G\to G$$ be associated morphisms satisfying the commutativity diagrams.
I wonder whether it is true that the binary operation $\otimes$ induced by $m$ is necessarily the binary operation $\circ$ given by the definition of group $G$.
To show this, I am considering using the Eckmann–Hilton argument, but I don't know how to show (and now I am not even sure) that $$(a \otimes b) \circ (c \otimes d) = (a \circ c) \otimes (b \circ d)$$
by playing with the commutative diagrams (?).
Thanks in advance!