reading martin's note ...

???so let's consider ... ??functor f : _set_^op X _set_^op -> _set_ which is continuous in each variable separately ... ????... ??using _set_^op as free complete on 1 object, seems pretty clear that category of such functors f is ess category of sets ??? ... ???which is pretty much what we were expecting ?? ....

?? so suppose that x and y are symmetric monoidal cocomplete v-enriched categories for some nice v ...

then consider tensor product of x and y as cocomplete v-enriched categories ... "universal bi-v-cocontinuous ..." ....

x#y # x#y -> x#y

....

let's try another approach for the moment ...

x symmetric monoidal category ... (secretly x = (2,1)-category of v-cocomplete v-enriched categories ... ???)

x' category of commutative monoids in x ...

x' inherits tensor product operation from x ... but in x' it's actually now the coproduct .... ????why ??? ....

m1,m2 commutative monoids in x ...

m1 # m2 gets commutative monoid structure how ????......

(m1 # m2) # (m1 # m2) -> (m1 # m2) .... ??coming from:

(m1 # m2) # (m1 # m2) -> (m1 # m1) # (m2 # m2) -> m1 # m2


??so ... martin's "F" and "G" live in m1 # m2 ... ??? ....

?? "m1 # m2" here is the v-enriched bifunctors from m1^op X m2^op to v, v-continuous in each variable separately ??? ....

?? so then first we should cook up the v-enriched quadrifunctor from m1^op X m1^op X m2^op X m2^op to v, v-continuous in each variable separately, taking (a,b,c,d) to F(a,c) # G(b,d) .... ????