TITLE: Distributivity of categorical product and sum
QUESTION [10 upvotes]: Let $\textbf{C}$ be a category with products and co-products. 
If $\textbf{C}$ is $\textbf{Set}$ or $\textbf{Top}$ I'm pretty sure the following "distributive law" holds: 
for all $X,Y,Z \in \operatorname{ob}(\textbf{C})$, $$(X \sqcup Y) \times Z \cong (X \times Z) \sqcup (Y \times Z).$$
Does this (or a similar law) hold for any such category $\textbf{C}$ or not?

REPLY [12 votes]: A simple counterexample is $\text{Set}^{op}$. Asking for $\text{Set}^{op}$ to be distributive means asking for coproducts to distribute over products in $\text{Set}$, which is easy to rule out for finite sets using cardinality. Another simple counterexample is $\text{Vect}$, where products and coproducts agree, and where it's again easy to rule out for finite-dimensional vector spaces using dimension. (However, it is true that tensor products distribute over direct sums; $\text{Vect}$ is not cartesian closed but it is the next best thing, closed monoidal.)
An interesting example is $\text{CRing}^{op}$, the category of affine schemes. It's a nice exercise to show that this category is distributive, despite neither being cartesian closed nor (as far as I know) admitting a conservative functor to $\text{Set}$ preserving products and coproducts (taking the Zariski spectrum is not conservative and does not preserve products).