Constructing a counterexample in category theory 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 $A \times B$ to mean the direct product and $A+$B to be the direct sum), whether there is an isomorphism between $A \times (B+C)$ and $(A\times B)+(A \times C)$.
An earlier question asked you to show that associativity of products holds. $(A \times B) \times C$ is isomorphic to $A \times (B \times C)$. This is true, even when not all objects have products (i.e. it is true that if  $A \times (B \times C)$ and $(A \times B) \times C$ 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 $A,B,C,A \times B,A \times,C,B+C,A\times (B+C),(A\times B)+(A\times C)$ and drew in all the arrows I know exist from the definitions of products and sums (e.g. $A \times B$ guarantees that there is an arrow from $A \times B$ to $A$ and one to $B$...). Then I observed that this diagram has no arrows that go from $A\times (B+C)$ to $(A\times B)+(A \times C)$ 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 $A\times (B+C)$ is not isomorphic to $(A\times B)+(A \times C)$, 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.

The dual to the statement that $A\times(B+C)$ is isomorphic to
$(A\times B)+(A\times C)$ is that $A+(B\times C)$ is isomorphic
to $(A+B)\times(A+C)$. Now if this statement fails in some
category $\mathcal{C}$ then the original fails in the opposite
category $\mathcal{C}^{\mathrm{op}}$.
