Proof
First observe that for two presheaves, their Cartesian product is a colimit over presheaves represented by Cartesian products in . Explicity, using coend-notation, we have:
(1)
where denotes the Yoneda embedding.
This is due to the following sequence of natural isomorphisms:
where the first step expands out both presheaves as colimits of representables separately, via the co-Yoneda lemma, the second step uses that the Cartesian product of presheaves is a two-variable left adjoint (by the symmetric closed monoidal structure on presheaves) and as such preserves colimits (in particular coends) in each variable separately, and under the brace we use the defining universal property of the Cartesian products, assumed to exist in .
Now observe that the colimit of a representable presheaf is the singleton.
(2)
One way to see this is to regard the colimit as the left Kan extension along the unique functor to the terminal category. By the formula there, this yields
where we made explicit the constant functor , constant on the singleton set , and then applied the co-Yoneda lemma.
Using this, we compute for the following sequence of natural isomorphisms:
Here the first step is (1), the second uses that colimits commute with colimits, the third uses again that the Cartesian product respects colimits in each variable separately, the fourth is (2), the last step is again the respect for colimits of the Cartesian product in each variable separately.