sketch is looking very sketchy at this point :/

sketch.org

@@ -22,13 +22,28 @@ arrows in D. Subject to the following equations:
     (p □ id) o (id □ g) = (id □ g) o (p □ id)
     (id □ q) o (f □ id) = (f □ id) o (id □ q)
 where p,q are tight morphisms.
+The tight morphisms are the (p □ q), i.e (p □ id) o (id □ q) or the
+equivalent flipped version.
 The monoids with respect to □ are strict premonoidal categories with a
 specified subset of the central morphisms.
 Restricted to tight arrows, it is the cartesian produt.
 
-There should be a free monoid monad T on M-category, and then "freyd
-multicategories" should be T-multicategories.
-What does this free monoid look like?
+** ... wrt Funny Cartesian Product of M-Cats
+
+We want a monad T that freely constructs an M-cat such that
+
+1. (TC)_tight is a cartesian category
+2. TC is a monoid with respect to the □-monoidal structure on M-cat.
+
+So we define (TC)t to be the free cartesian category on Ct.
+TC_l is generated by
+
+1. For every p ∈ (TC)t, a morphism (ret p) ∈ (TC)l
+2. For every A₁,...,Aᵢ,...,Aₙ and morphism p : Ai -> B, a morphism
+   \[ let i = p \]
+3. Satisfying hm...
+
+** Freyd Multicategories as T-multicategories
 
 Let's spell this structure out and then see how to simplify.
 A freyd multicategory consists of
@@ -39,9 +54,11 @@ A freyd multicategory consists of
    consists of
    - For every list Γ ∈ C0* and output type A ∈ C0, a set of pure
      morphisms C1ᵥ(Γ;A) and effectful morphisms C1ₜ(Γ;A) and an
-     injective function i : C1ᵥ(Γ;A) -> C1ₜ(Γ;A)
+     injective function ret : C1ᵥ(Γ;A) -> C1ₜ(Γ;A)
    - For every object A, a pure identity arrow id(A) : C1ᵥ(A;A)
-   - TODO: the rest
+   - For every let-sequence T(C1ₜ)ₜ(Δ1,...,Δn ; Γ) and term C1ₜ(Γ;A),
+     a term C1ₜ(Δ1,...,Δn;A)
+   - 
 
 These have a set C0 of objects, for every list of objects G and object
 A a set of loose morphisms C_l(G;A) with a specified subset of tight