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.
 A: Yes, and here is a proof which is entirely formal (or rather, offloads the work onto understanding Day convolution).
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 $\Delta$.
Therefore colim takes monoids (lax monoidal functors) to monoids.
