nCatLab states that a monoid in the Day-convolution monoidal category is equivalent to a lax monoidal functor. In the Haskell community this result is used to explain the construction of the applicative functor. Rivas and Jaskelioff show that a monoid in the endofunctor category $[Set, Set]$ with Day convolution is equivalent to a *strong* lax monoidal functor. They need strength to get the natural transformation $Id \to F$ from the (lax) preservation of the unit $1 \to F(1)$. I'm trying to understand why they need strength even though the nCatLab result doesn't. Granted, nCatLab deals with enriched functors, and self-enrichment is related to strength, but functors in $Set$ are both strong and self-enriched.

Instead of starting from $[Set Set]$, I'd like to start from a monoidal category $C$ and work with the functor category $[C, Set]$ with Day convolution. A monoid is a functor $F$ in this category, together with two natural transformations. I'm interested in one of them.

The unit under Day convolution is $C(I, -)$ where $I$ is the unit object in $C$. Consider the set of natural transformations from $C(I, -)$ to $F$:

$[C, Set](C(I), -), F) = \int_a C(I, a) \to F(a)$

By Yoneda, this is equal to $F(I)$.

A function from $1$, the singleton set, will pick up a natural transformation in $[C, Set](C(I, -), F)$ and an element of $F(I)$:

$1 \to F(I)$

This is exactly one of the morphisms defining the lax monoidal functor. It seems like, in this formulation, strength is not needed. (It's not needed in the rest of the proof either.) What am I missing?

endofunctorof a monoidal category. $\endgroup$ – Mike Shulman Feb 1 '17 at 0:27