sequential limits

A category has sequential limits if it has limits of diagrams of the following form: $\cdots \bullet \to \bullet \to \bullet$

equalizers and countable products implies sequential limits

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 products and sequential limits implies countable products

If $X_1,X_2,\dotsc$ is an infinite sequence of objects, then their product is the limit of the sequence $\cdots \to X_2 \times X_1 \to X_1$ .
filtered limits implies sequential limits

trivial
self-dual and sequential limits implies sequential colimits

trivial by self-duality
self-dual and sequential colimits implies sequential limits

trivial by self-duality

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