countable coproducts

A category has countable coproducts if it has coproducts for countable families of objects.

coproducts implies finite coproducts and countable coproducts

[dualized] trivial
countable coproducts implies finite coproducts

[dualized] trivial
coequalizers and countable coproducts implies sequential colimits

[dualized] Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case. Namely, the limit of $\cdots \to X_2 \to X_1 \to X_0$ is the equalizer of two suitable endomorphisms of $\prod_{n \geq 0} X_n$ .
finite coproducts and sequential colimits implies countable coproducts

[dualized] If $X_1,X_2,\dotsc$ is an infinite sequence of objects, then their product is the limit of the sequence $\cdots \to X_2 \times X_1 \to X_1$ .
self-dual and countable products implies countable coproducts

trivial by self-duality
self-dual and countable coproducts implies countable products

trivial by self-duality

For these categories the database has no info if they satisfy this property or not.

—