A small set of prefixes recurs across category theory, each a fixed modification of functor, monad, or monoidal structure. Collected in one table they read as a grammar rather than as scattered jargon.
| prefix | attaches to | modification | instances |
|---|---|---|---|
| pro- | functor, monad | a two-sided generalisation through | profunctor, promonad, produoidal category |
| pre- | sheaf, monoidal | the structure before a coherence condition is imposed | presheaf (before gluing), premonoidal (before interchange) |
| para- | category | morphisms carry an explicit parameter object | $\mathrm{Para}$ |
| pra- | functor | parametric right adjoint, a right adjoint after slicing | polynomial functors, the universe row of contextads |
| co- | any | the categorical dual, every arrow reversed | comonad, colimit, coend, copresheaf |
| lax / colax | functor, action | structure preserved only up to a fixed, non-invertible comparison map | lax monoidal functor, colax action |
Three of these carry a subtlety worth stating on its own.
The pre- prefix names an absence. A presheaf is a functor into sets with no gluing axiom, so it is a sheaf before descent, whereas an premonoidal category has a tensor functorial in each argument separately but with no interchange law, so it is monoidal before the two orders of a side effect are forced to agree.
The pra- prefix is the least familiar and the most load-bearing for the polynomial world. A parametric right adjoint is a functor that becomes a right adjoint once its domain is sliced over an object, which is the property that makes polynomial functors composable and gives the representable rows of the Ctx construction their universe reading.
Lax and colax record the direction of the surviving comparison. A lax functor supplies a map from the image of a composite into the composite of the images, a colax functor the reverse, and the colax actions behind the Ctx construction are the reason the distinction is not cosmetic.
This table is a Rosetta Stone for terminology rather than for a mathematical object, aligning the prefixes across the constructions they modify.