Dear all,

Given a V-enriched category C and sufficiently many assumptions on V and C, there is an equivalence between (symmetric) lax monoidal functors C --> V and (commutative) monoids in Fun(C,V) with respect to Day convolution. I am interested in knowing if there is a similar characterization in the literature for strong monoidal functors, as a property of objects of Mon(Fun(C,V)) or possibly BiMon(Fun(C,V)). The latter contains functors with a monoid structure for Day convolution and a comonoid structure for Day coconvolution, obtained by taking a right Kan extension in place of the usual left Kan extension in the definition of Day convolution.

Best,
Lorenzo