# Commit 2021-12-03 16:11 f0a1cd1a

View on Github →feat(category_theory/*): Representably flat functors (#9519)
Defined representably flat functors as functors `F : C ⥤ D`

such that `(X/F)`

is cofiltered for each `X : C`

.

- Proved that if
`C`

has all finite limits and`F`

preserves them, then`F`

is flat. - Proved that flat functors preserves finite limits.
In particular, if
`C`

is finitely complete, then flat iff left exact. - Proved that if
`C, D`

are small, then`F`

flat implies`Lan F.op`

preserves finite limits implies`F`

preserves finite limits. In particular, if`C`

is finitely cocomplete, then flat iff`Lan F.op`

is left exact.