# 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 $\phi$?

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In the context of your question, there are no things in $(A\times B)\oplus(A\times C)$. – Mariano Suárez-Alvarez Sep 16 '11 at 22:35
Yes, I'm still not very sure of the language to use while communicating my problems here. – groops Sep 16 '11 at 22:43
Even Wikipedia contains the answer. Shame on you. – beroal Sep 17 '11 at 12:16
@beroal Where? I didn't see it mentioned in the article on distributive categories. – groops Sep 17 '11 at 23:58
@groops: en.wikipedia.org/wiki/Distributive_category $[1\times\iota_1,1\times\iota_2]$ – beroal Sep 27 '11 at 12:34

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.