Let \(\mathcal C\) be a category and \((X_\alpha)_{\alpha\in A}\) a family of objects of \(\mathcal C\). A coproduct of \((X_\alpha)_{\alpha\in A}\) is an object \(\coprod_{\alpha\in A} X_\alpha\) together with morphisms \(\iota_\beta\colon X_\beta\to\coprod_{\alpha\in A} X_\alpha\) for \(\beta\in A\) such that for every object \(Y\) and every family of morphisms \(f_\beta\colon X_\beta\to Y\) there is a unique morphism \(f\colon\coprod_{\alpha\in A} X_\alpha\to Y\) with \(f\circ\iota_\beta = f_\beta\) for every \(\beta\in A\).
The coproduct in \(\mathcal C\) is the product in the opposite category \(\mathcal C^{\mathrm{op}}\). By the uniqueness in the definition, two coproducts of the same family are isomorphic by a unique isomorphism compatible with the morphisms \(\iota_\beta\).
In \(\Top\) the coproduct is the disjoint union \(\coprod_\alpha X_\alpha\), a subset \(U\) being open if \(U\cap X_\alpha\) is open in \(X_\alpha\) for every \(\alpha\). In the category of based spaces it is the wedge sum \(\bigvee_\alpha X_\alpha\). In \(\Ab\) it is the direct sum \(\bigoplus_\alpha X_\alpha\), and in \(\Grp\) it is the free product \(\ast_\alpha X_\alpha\).
See [@Hat02].