# Commutation of filtered colimits with pullbacks

It is known that in a locally $\lambda$-presentable category $\lambda$-filtered colimits commute with $\lambda$-small limits, and hence with pullbacks. I guess that this fails for $\mu$-filtered colimits for $\mu<\lambda$?

Thus, I am looking for

• a regular cardinal $\lambda$
• a locally $\lambda$-presentable category $\mathcal{C}$, which I would like to require to be additive,
• an ordinal $\mu < \lambda$
• a diagram $(X_{\alpha})_{\alpha<\mu}$ in $\mathcal{C}$,
• a morphism $Y \to S$ in $\mathcal{C}$
• compatible morphisms $X_{\alpha} \to S$ in $\mathcal{C}$,

such that the canonical morphism

$$\varinjlim_{\alpha<\mu} (X_{\alpha} \times_S Y) \to \bigl(\,\varinjlim_{\alpha<\mu} X_{\alpha}\,\bigr) \times_S Y$$

is not an isomorphism. Can you give such an example?

• Why the downvote? – HeinrichD May 23 '17 at 19:22
• Why is $\cal C$ additive? – Fosco May 23 '17 at 20:16
• @FoscoLoregian: I would like to require this, because currently I work with additive categories. Hence, a counterexample should not come from non-additivity. – HeinrichD May 24 '17 at 6:04
• The first version of this question was about products, but they are just coproducts in the additive case, so that the question was trivial. I edited the question - it is now about pullbacks, which is the situation which I am actually interested in. – HeinrichD May 24 '17 at 6:23