# What is the canonical morphism in a category where finite products and coproducts exist?

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 $\phi$?

#### Solutions Collecting From Web of "What is the canonical morphism in a category where finite products and coproducts exist?"

There is a canonical morphism $B\to B\oplus C$, which induces a morphism $\phi_1:A\times B\to A\times(B\oplus C)$. Similarly, there is a canonical morphism $C\to B\oplus C$ which induces a morphism $\phi_2:A\times C\to A\times(B\oplus C)$.

Now $\phi_1$ and $\phi_2$ determine a unique morphism $(A\times B)\oplus(A\times C)\to A\times(B\oplus C)$. That’s your morphism.