Implication Details

Claim: If a category has countable products and has ℵ₁-cofiltered limits, then it has products.

Proof: The product $\prod_{i \in I} X_i$ is the $\aleph_1$-cofiltered limit of the countable partial products $\prod_{i \in C} X_i$ where $C$ ranges over the countable subsets of $I$.