Been reading up on the idea of distributive categories. Suppose $\mathcal{C}$ is some category such that for all $A,B\in\mathcal{C}$ the product $A\times B$ and coproduct $A\oplus B$ exist.
So $\mathcal{C}$ is a distributive category if the canonical morphism $ \phi\colon (A\times B)\oplus(A\times C)\to A\times (B\oplus C) $ is an isomorphism.
This is a basic question, but what precisely is this so called canonical morphism? Really, what does an arbitrary "thing" (not sure if element is the right word here) in $(A\times B)\oplus(A\times C)$ look like, and where does it go under $\phi$?