final class Zygo[F[_], S, A, B](val aux: F[B] =>B, val alg: F[(B, A)] =>A)(using F: Traverse[F], P: Project[F, S]) extends ReadScheme[S, A]
Zygomorphism citizen — a fold carrying an auxiliary algebra alongside the main one (ReadScheme) with X = F[(B, A)]: each child slot pairs the auxiliary result B with the main result A.
aux: F[B] => B runs a second, self-contained fold whose results the main alg: F[(B, A)] => A may read per child. It is the rung the comonad tower skips between Cata (X = Nothing) and Para (X = F[(S, A)]): para is exactly zygo at B = S with aux = embed (the auxiliary fold rebuilds the original subterm), and ignoring the B half (alg ∘ map(_._2)) degenerates to Cata. The further generalisation — letting aux also see the A half — is the mutumorphism (Mutu).
'''On the existential.''' X = F[(B, A)] is the store comonad over the auxiliary carrier B, the same store-comonad complement Para flags as its writable candidate, but over an arbitrary B rather than the structure S. The two results are computed in one pass (the fold yields (B, A) pairs; the final projection keeps the A). Stack-safe (the Machines.foldLayered machine).
Build-then-observe across the build-output ⇄ read-input seam, preserving structure, on a shared carrierF. Flip self (it must be reversible — Accessor[F]andReverseAccessor[F], i.e. an Iso or Review over Direct) so it reads T from B, then andThenthat under the same carrier. The result is the full Optic[B, A, C, D, F], not a collapsed getter: its read capability follows the carrier (.get for Direct), and self's read focus A survives as the composite's write-back focus.
Build-then-observe across the build-output ⇄ read-input seam, preserving structure, on a shared carrierF. Flip self (it must be reversible — Accessor[F]andReverseAccessor[F], i.e. an Iso or Review over Direct) so it reads T from B, then andThenthat under the same carrier. The result is the full Optic[B, A, C, D, F], not a collapsed getter: its read capability follows the carrier (.get for Direct), and self's read focus A survives as the composite's write-back focus.
This is exactly self.reverse.andThen(that). The motivating case is ana.cross(cata): a Review (the unfold) crossed with a getter on the built S (the fold) — a (materializing) hylomorphism whose .get reads the folded value. When that sits on a different carrier (a Prism, a Fold, …), the cross-carrier cross overload below is selected instead.
Seam: that's source is self's T and its back-type is self's S.
Existential leftover carried alongside the focus — the type-level witness the carrier uses to rebuild T. Concrete at construction (Lens.apply sets X = S, Prism.apply sets X = S, …) and abstract when the optic is bound to Optic[…, F] without refinement.
Existential leftover carried alongside the focus — the type-level witness the carrier uses to rebuild T. Concrete at construction (Lens.apply sets X = S, Prism.apply sets X = S, …) and abstract when the optic is bound to Optic[…, F] without refinement.
ANY outer ∘ read-only inner — the inner is honestly one-way (T = Unit: a Getter, AffineFold, or Fold), so only the two READ sides matter and the composite collapses to the read-only join of their strengths via compose.ReadCompose (Getter / PickFold / ForgetFold).
ANY outer ∘ read-only inner — the inner is honestly one-way (T = Unit: a Getter, AffineFold, or Fold), so only the two READ sides matter and the composite collapses to the read-only join of their strengths via compose.ReadCompose (Getter / PickFold / ForgetFold).
A trait member (not an extension in the companion) deliberately: once a receiver is statically one of the fused concrete classes, its andThen member overloads enter resolution and Scala 3 never falls back to extension methods when they all fail — the collapse must be in the member overload set to be reachable without an expected-type ascription.
Only the inner's T is pinned to Unit; its B stays free (IB) even though read-only inners always have B = Unit. That keeps this overload strictly LESS specific than the same-carrier andThen above (which accepts every argument this one does whenever B = Unit at the receiver), so a same-carrier read-only ∘ read-only call resolves unambiguously to the AssociativeFunctor path and this one fires exactly on the cross-seam cells the generic member cannot type.
Compose with another optic under the shared carrier F. Requires AssociativeFunctor[F]. Cross-carrier composition (Lens → Optional, Lens → Traversal, …) goes through the Morph-summoning overload of this same method.
Compose with another optic under the shared carrier F. Requires AssociativeFunctor[F]. Cross-carrier composition (Lens → Optional, Lens → Traversal, …) goes through the Morph-summoning overload of this same method.
Attributes
Example
case class Address(street: String)
case class Person(address: Address)
val streetLens = lens[Person](_.address).andThen(lens[Address](_.street))