Construction of the coproduct of two monoidal categories Consider the 2-category $\mathbf{MonCat}$ of monoidal categories with strong monoidal functors and monoidal natural transformations.  Where can I find an explicit construction of the coproduct $M + N$ of two monoidal categories $M$ and $N$?
By analogy with monoids, I would expect this to have objects words of objects on the two underlying monoidal categories. What are the morphisms, precisely?  I imagine the notion of coproduct in a 2-category requires some weakening of the conditions (is this the thing I am looking for?)
 A: $\def\M{\mathcal M}
\def\N{\mathcal N}
\require{AMScd}$
It seems to me that the following argument is correct. Please point out any flaw that you might find!
Let $(\M, \otimes_M, I), (\N, \otimes_N, J)$ the two monoidal categories.
Define a category $\M \star \N$ having objects the tuples
$$
(\underline M; \underline N)_r = M_1 \boxtimes N_1\boxtimes \cdots\boxtimes M_r \boxtimes N_r
$$ of length $r\ge 1$, and the object $I\boxtimes J$ as tentative unit; these objects are reduced according to the same rule of the free product of monoids. 
Given two tuples of objects, the hom-set between them is
$$
\M \star \N ((\underline M; \underline N)_r, (\underline M'; \underline N')_r) = \prod_i 
\M(M_i, M_i') \times \N(N_i, N_i')
$$
if the tuples have the same length, and it's empty otherwise.
With this definition on $\M \star \N$ given a diagram 
$$
\begin{CD}
\M @>U>> \mathcal A @<V<< \N
\end{CD}
$$ of strong monoidal functors, there is a unique functor $\langle U,V\rangle$ from $\M \star \N$ to $\mathcal A$, defined by sending the tuple $(\underline M; \underline N)_r$ to
$$
UM_1 \otimes_A VN_1\otimes_A \cdots\otimes_A UM_r \otimes_A VN_r
$$
and doing the same on morphisms. 
This is a strong monoidal functor, because $U,V$ are, and unique, because $\M\star \N$ is "generated" by dyads of the form $I\boxtimes N$ and $M\boxtimes J$, on which every two functors with the same property of factoring $U,V$ through $i_\M, i_\N$ must coincide. More in detail, assume $\langle U,V\rangle$ and $|U,V|$ are strong monoidal functors $\M \star \N \to \mathcal A$ such that 
$$
\begin{cases}
\langle U,V\rangle \circ i_\M = |U,V| \circ i_\M\\
\langle U,V\rangle \circ i_\N = |U,V| \circ i_\N
\end{cases}
$$ I claim that since this is true, they coincide everywhere. And this is true because given a tuple $ M_1 \boxtimes N_1\boxtimes \cdots\boxtimes M_r \boxtimes N_r$ I can "interpolate" $I\boxtimes J$ everywhere and use associativity of juxtaposition, to obtain that the tuple is equivalent to
$$
M_1 \boxtimes I\boxtimes J\boxtimes N_1\boxtimes I\boxtimes J\boxtimes\cdots\boxtimes M_r \boxtimes I\boxtimes J\boxtimes N_r
$$ now it suffices to apply $\langle U,V\rangle$ and $|U,V|$.
