How do I prove that canonical monomorphisms of a coproduct in the category of pointed spaces are topological embeddings? Let $\{(X_i,p_i)\}$ be a family of pointed spaces and $(\coprod X_i,j_i)$ be a coproduct of $\{(X_i,p_i)\}$ in the category of pointed spaces. 
I have proven that canonical monomorphisms $j_i$'s are indeed topological embeddings by observing a special coproduct, namely the wedge sum of $\{(X_i,p_i)\}$. However, I'm curious if there is a purely categorical way (i.e. by universal property) to prove that $j_i$'s are topological embeddings.
 A: For simplicity of notation I will talk only about binary coproducts, but everything I will say works for arbitrary coproducts.  Assume that you are in a category that has a map between any two objects (in particular, for instance, this is true in any category with a zero object, and the converse is true if you assume the category has both an initial object and a terminal object).  If you have a coproduct $X\stackrel{i}\to Z\stackrel{j}\leftarrow Y$, then there is a map $p:Z\to X$ such that $pi=1$ and $pj=f$, where $f$ is any map $X\to Y$.  It follows that the inclusion $i:X\to Z$ has a left inverse, i.e. it is a split monomorphism.  Split monomorphisms are pretty much the nicest kind of "embedding" you can have in any category.  In particular, for topological spaces, it is easy to see that any split monomorphism must be a topological embedding.  More generally, the topological embeddings are exactly the regular mononomorphisms in the category of topological spaces, and it is easy to see that a split monomorphism is regular in any category (namely, with the notation above, it is the equalizer of $1:Z\to Z$ and $ip:Z\to Z$).
