In MacLane's Categories for the Working Mathematician, page 110, second edition, he states that, in the category of groups $Grp$, $F_n$ being groups and $x_n, y_n$ being two cones on these groups (thus a family of group morphisms from X (resp. Y) to $F_n$), these cones form a group by multiplication $x_n y_n$.

As I understand it, it just seems false. Let be two group homomorphisms $g, f$, there is no reason why $x \mapsto g(x) f(x)$ should also be a group morphism, as we cannot conclude whether $g(x)f(x)g(x)^{-1}f(x)^{-1} = e$.

So I probably missed something... I could use a hint!