Suppose $\mathcal{C}$ is a category with small colimits, and $G \in \mathrm{Ob}(\mathcal{C})$ is a strong generator. (This means that $f: X \to Y$ is an isomorphism iff $f\_{\*}: \mathrm{Hom}(G,X) \to \mathrm{Hom}(G,Y)$ is a bijection.)



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> How does one prove that every object $X$ is a colimit of copies of $G$ (i.e., a quotient of coproducts of $G$)?
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I think $X \simeq \mathop{\mathrm{colim}}\_{G \to X} G$. I can see there is a natural map $e: \mathop{\mathrm{colim}}\_{G \to X} \to X$, and that the induced map $e\_{\*}: Hom(G,\mathop{\mathrm{colim}}\_{G \to X}) \to Hom(G, X)$ is surjective, but I cannot see that it map is injective.



Proof that objects are colimits of generators
I think there are actually *three* possible things that you might be asking, but the answer to all of them is no. Suppose that G is a strong generator in a cocomplete category C. Then you can ask:


1. Is every object X of C the colimit of G over the canonical diagram of shape $(G\downarrow X)$? (If so, then G is called *dense* in C.)
2. Is every object *some* colimit of a diagram all of whose vertices are G? (If so, then G is called *colimit-dense* in C.)
3. Is C the smallest subcategory of itself containing G and closed under colimits? (If so, then G is a *colimit-generator* of C.)


The category of compact Hausdorff spaces is a counterexample to the first two. It is monadic over Set (the monad is the ultrafilter monad, aka Stone-Cech compactification of the discrete topology), and hence cocomplete, and the one-point space is a strong generator. But any colimit of a diagram consisting entirely of 1-point spaces must be in the image of the free functor from Set (the one-point space being its own Stone-Cech compactification), and hence (since that functor is a left adjoint and preserves colimits) must be the free object on some set. However, not every compact Hausdorff space is the Stone-Cech compactification of a discrete set.


As Todd pointed out in the comments, though, CptHaus is the colimit-closure of the one-point space, since every object is a coequalizer of maps between free ones (because the category is monadic over Set); thus it isn't a counterexample to the third question. Counterexamples to the third question are actually much harder to come by, and in fact if you assume additionally that C has finite limits and is "extremally well-copowered", then it is true that any strong generator is a colimit-generator. I think there's a proof of this somewhere in Kelly's book "Basic concepts of enriched category theory," and a brief version can be found [here](https://home.sandiego.edu/~shulman/papers/generators.pdf). However, without these assumptions, one can cook up ugly and contrived counterexamples, such as example 4.3 in the paper "Total categories and solid functors" by Borger and Tholen.


