Heyting-Lewis Logic is the extension of intuitionistic propositional logic with a strict implication connective that satisfies the constructive counterparts of axioms for strict implication provable in classical modal logics. Variants of this logic are surprisingly widespread: they appear as Curry-Howard correspondents of (simple type theory extended with) Haskell-style arrows, in preservativity logic of Heyting arithmetic, in the proof theory of guarded (co)recursion, and in the generalization of intuitionistic epistemic logic.Heyting-Lewis Logic can be interpreted in intuitionistic Kripke frames extended with a binary relation to account for strict implication. We use this semantics to define descriptive frames (generalisations of Esakia spaces), and establish a categorical duality between the algebraic interpretation and the frame semantics. We then adapt a transformation by Wolter and Zakharyaschev to translate Heyting-Lewis Logic to classical modal logic with two unary operators. This allows us to prove a Blok-Esakia theorem that we then use to obtain both known and new canonicity and correspondence theorems, and the finite model property and decidability for a large family of Heyting-Lewis logics.1 Curiously, Lewis was not using as a primitive, so in fact his intuitionistically problematic definition of ϕ ψ was ¬ (ϕ ∧ ¬ψ). See [71, App. D] for an account of problems caused by Lewis' use of a Boolean propositional base, namely trivialization [64, 67] of his original system [63, 66], which in turn finally lead him to propose systems S1-S3 [68, App. 2] as successive "lines of retreat" [88]. Lewis considered S4 and S5, suggested by Becker [8], too strong to provide a proper account of strict implication [68, p. 502] and appeared frustrated with later development of modal logic [71, § 2.1]. Yet, despite his supportive attitude towards non-classical logics, he seems to have mentioned Brouwer only once (favourably) [65], and does not appear to have ever referred to, or even be familiar with subsequent work of Kolmogorov, Heyting or Glivenko [71, § 2.2].7 To the best of our knowledge, the only explicit (if brief) discussion of the Curry-Howard connection between Haskell arrows and i-Sa − is found in Litak and Visser [71, § 7.1].8 The logical perspective seems to cast a light on the controversy whether arrows are "stronger" than applicative functors [70,77]. Putting aside the general question of whether one takes as a measure of strength the capability to inhabit more types or rather to allow more distinctions, in the presence of , the situation is not as clear-cut as in the unary case, where (monadic) PLL simply extends (applicative) IntK ⊕ S . Defining arrows over the latter set of axioms via delay yields i-Sa − ⊕ Box, whereas inhabiting apply with Appa yields PLAA. These are two incomparable systems.