# Colimit of a lax monoidal functor is a monoid

The question here is: Given any lax monoidal functor $$F:C \to V$$, where $$C$$ is a small monoidal category and $$V$$ is a cocomplete monoidal category where the tensor product distributes over colimits in both variables (i.e. tensoring with any object on either side is cocontinuous), is the colimit of $$F$$ always a monoid in $$V$$?

Attempt

If $$C$$ has a terminal object $$1$$, then it is known that the colimit of any functor with domain $$C$$ is simply its evaluation at $$1$$. Since $$1$$ is trivially a monoid in $$C$$ and lax monoidal functors send monoids to monoids, it follows that the colimit $$F(1)$$ is a monoid in $$V$$ in this case.

Now, consider the general case. Since $$F$$ is lax monoidal, we have a map $$I_V \to F(I_C)$$, where $$I_C$$ and $$I_V$$ are the unit objects of $$C$$ and $$V$$ respectively. Composing this map with the component of the colimiting cocone at $$I_C$$ then gives a map $$I_V \to \varinjlim F$$ in $$V$$ that ought to be the unit of a monoid.

It is now time to consider the multiplication. Since the tensor product distributes over colimits, we have an isomorphism between $$\varinjlim F \otimes \varinjlim F$$ and the colimit of the functor $$C \times C \to V$$ that sends $$(A_1, A_2)$$ to $$F(A_1) \otimes F(A_2)$$. Hence, to define a multiplication on $$\varinjlim F$$, it suffices to define a compatible family of maps $$F(A_1) \otimes F(A_2) \to \varinjlim F$$. The idea here is to compose the map $$F(A_1) \otimes F(A_2) \to F(A_1 \otimes A_2)$$ (which exists because $$F$$ is lax monoidal) with the component of the colimiting cocone at $$A_1 \otimes A_2$$ to get the components of the required compatible family. This compatible family then induces a map $$\varinjlim F \otimes \varinjlim F \to \varinjlim F$$ that ought to be the multiplication of a monoid with the map from the last paragraph as the unit.

Proving associativity and unitality in the colimit monoid requires using the coherence axioms for $$F$$, and is omitted here.

In fact, we can also show that any monoidal natural transformation between two lax monoidal functors $$C \to V$$ induces a morphism of monoids between the colimits, so we actually have a functor $$Lax(C, V) \to Mon(V)$$ from lax monoidal functors to monoids.

If C and V are monoidal, $$\text{Fun}(C,V)$$ inherits a monoidal structure via Day convolution. Monoids in $$\text{Fun}(C,V)$$ are lax monoidal functors. See nlab on Day convolution for details.
It will suffice to prove that the functor $$\text{colim}:\text{Fun}(C,V)\rightarrow V$$ is lax symmetric monoidal, as then it would take monoids to monoids: $$\text{colim}:\text{Fun}^\text{lax}(C,V)\rightarrow\text{Mon}(V).$$ To do so, notice that the diagonal functor $$\Delta:V\rightarrow\text{Fun}(C,V)$$ is oplax monoidal, via the structural maps $$v\otimes w\rightarrow v\otimes_\text{Day} w$$ where $$v$$, $$w$$, and $$v\otimes w$$ are the constant functors.
By doctrinal adjunction, giving an oplax monoidal structure on a functor is equivalent to giving a lax monoidal structure on its left adjoint. Hence we have just described a lax monoidal structure on the functor "colim", which is left adjoint to $$\Delta$$.