I $\{X_i\}$ is a small family of associative $\mathbb{C}$-algebras and $X$ is their free product.
Then I have two questions:
1) Why is $X$ their coproduct?
2) Is the Hochschild homology of X isomorphic to the direct sum of the Hochschild homologies of the $X_i$?
IN both cases why or why not?
Thanks in advance.