general topology - identification of the wedge sum with subset of the cartesian product

Let $(X,x_0),(Y,y_0),(Z,z_0)$ be based spaces.
Define the wedge sum
$X\vee Y\vee Z$ to be $X\sqcup Y\sqcup Z$ modulo $x_0\sim y_0\sim z_0$.
How does this wedge sum relates to the subspace $$S=\{(x,y,z)\in X\times Y\times Z \;|\; \text{only one of the entries is not the base point} \}$$

I Know they are identified but i'm not sure how to prove it. Let
$$f:X\sqcup Y\sqcup Z\to S;\; f(x)=(x,y_0,z_0), f(y)=(x_0,y,z_0), f(z)=(x_0,y_0,z)$$ Then obviously $f$ is surjective and such that
$f(x_0)=f(y_0)=f(z_0)$ hence $f$ facotors through $X\vee Y\vee Z$ to give a homeomorphism. Is this argument correct and especially is $f$ continuous?