Remarks:

(1) The construction can be carried out in exactly the same manner in the categories Gr and Top. In the Category Gr, the coproduct $\tilde G$ of the groups $G_{\alpha}$ is the free product with the group operation written multiplicatively and the candidate for $N$ is the normal subgroup generated by

$$\{x_{\alpha}x_{\beta}^{-1}\;/ x_{\alpha} \sim x_{\beta}\},$$

where as before we regard each $G_{\alpha}$ to be a subgroup of $\tilde G$ to simplify notations.

(2) In the category Top we proceed analogously by taking the coproduct, the disjoint union of the spaces, and defining the equivalence relation (40.4) on it and passing on to the quotient space. In applications one uses the defining properties (1) and (2) of definition (40.2) and not these details involved in the actual construction.