Exercise 10 in Geroch's Mathematical Physics asks whether direct products distribute over direct sums in arbitrary categories. (They do in the category of sets, which is what motivates the question). That is, (using AxB to mean the direct product and A+B to be the direct sum), whether there is an isomorphism between Ax(B+C) and (AxB)+(AxC).
An earlier question asked you to show that associativity of products holds. (AxB)xC is isomorphic to Ax(BxC). This is true, even when not all objects have products (i.e. it is true that if both Ax(BxC) and (AxB)xC exist, then they are isomorphic...). So I take this question to not require that all objects have products...
I don't believe there is the relevant isomorphism, but I'm not sure what I've done proves it. So my question is: Is the following argument a good way of approaching category theory questions?
So I wrote out a A,B,C,AxB,AxC,B+C,Ax(B+C),(AxB)+(AxC) and drew in all the arrows I know exist from the definitions of products and sums (e.g. AxB guarantees that there is an arrow from AxB to A and one to B...). Then I observed that this diagram has no arrows that go from Ax(B+C) to (AxB)+(AxC) or the other way. That is, even allowing for composition of arrows, there need not be a morphism between the two above mentioned objects. So, I said, this means that this diagram is a diagram of a category where Ax(B+C) is not isomorphic to (AxB)+(AxC), since if there are no morphisms between them, there can't be any isomorphisms.
Is this a good way of constructing counterexamples in category theory? Is there any rigorous discussion of "diagrams" used this way? (I've heard it mentioned, but I don't know where to look).
This is a question about whether the strategy I am using is a good one, not really about the actual truth value of the statement in question.