Implication Details

Claim: If a category has countable copowers and has ℵ₁-filtered colimits, then it has copowers.

Proof: Let $\C$ be a category with $\aleph_1$-filtered colimits and countable copowers. Let $X \in \C$ be an object and $I$ be a set. The poset $P_{<\aleph_1}(I)$ of countable subsets of $I$ is $\aleph_1$-filtered, and we have a diagram $P_{<\aleph_1}(I) \to \C$, $A \mapsto A \otimes X$. Its colimit is the copower $I \otimes X$.