Let $\mathbf{SMC}$ be the $2$-category of symmetric monoidal categories and strong symmetric monoidal functors.
Also, let $\mathbf{SMC}_{0}$ be the full sub-$2$-category of $\mathbf{SMC}$ on the symmetric monoidal categories whose unit object is initial (also called "semicocartesian").
Then, the inclusion $\mathbf{SMC}_{0} \hookrightarrow \mathbf{SMC}$ has a right $2$-adjoint mapping each symmetric monoidal category $(\mathbf{C},\otimes,I)$ to the coslice category $I/\mathbf{C}$ with the obvious tensor product induced by the isomorphism $I \cong I \otimes I$.
Now, given any symmetric monoidal category $(\mathbf{C},\otimes,I)$, define a new category $\mathbf{\bar{C}}$ and a functor $F:\mathbf{C} \to \mathbf{\bar{C}}$ as follows:
- The objects of $\mathbf{\bar{C}}$ are those of $\mathbf{C}$.
- The morphisms from $X$ to $Y$ in $\mathbf{\bar{C}}$ are the connected components of the comma category $- \otimes X \downarrow Y$ (or equivalently, the colimit of $\mathrm{Hom}_{\mathbf{C}}(- \otimes X,Y):{\mathbf{C}}^{\mathrm{op}} \to \mathbf{Set}$).
- The identity on $X$ is the component containing $(I, \lambda_{X}:I \otimes X \cong X)$.
- The composition of the components containing $(A, f:A \otimes X \to Y)$ and $(B, g:B \otimes Y \to Z)$ is the component containing $(B \otimes A, g \circ (1_B \otimes f) \circ \alpha)$ (where $\alpha$ is the associativity isomorphism $(B \otimes A) \otimes X \cong B \otimes (A \otimes X)$).
- The functor $F$ maps each object to itself and each morphism $f:X \to Y$ to the component containing $(I,f \circ \lambda_X:I \otimes X \to Y)$.
Then, define a symmetric monoidal structure on $\mathbf{\bar{C}}$ as follows:
- The tensor product of two objects of $\mathbf{\bar{C}}$ is their tensor product in $\mathbf{C}$.
- The tensor product of two morphisms $[(A_1, f_1:A_1 \otimes X_1 \to Y_1)]:X_1 \to Y_1$ and $[(A_2, f_2:A_2 \otimes X_2 \to Y_2)]:X_2 \to Y_2$ in $\mathbf{\bar{C}}$ is $[(A_1 \otimes A_2, (f_1 \otimes f_2) \circ \sigma)]:X_1 \otimes X_2 \to Y_1 \otimes Y_2$ (where $\sigma:(A_1 \otimes A_2) \otimes (X_1 \otimes X_2) \to (A_1 \otimes X_1) \otimes (A_2 \otimes X_2)$ is the "middle" isomorphism induced by the symmetric monoidal structure).
- The structural isomorphisms in $\mathbf{\bar{C}}$ are the images of the ones in $\mathbf{C}$ under the functor $F$.
It is easily seen that $\mathbf{\bar{C}}$ is semicocartesian, as the comma category $- \otimes I \downarrow Y$ is just the slice category $\mathbf{C}/Y$, which has a terminal object; and any category with a terminal object is connected.
Now, given any strong symmetric monoidal functor $G:\mathbf{C} \to \mathbf{D}$ where $\mathbf{D}$ is semicocartesian, the unique "extension" $\bar{G}:\mathbf{\bar{C}} \to \mathbf{D}$ should map objects the same way that $G$ does and each morphism $[(A, f:A \otimes_C X \to Y)]:X \to Y$ to the composition $G(X) \cong I_D \otimes_D G(X) \to G(A) \otimes_D G(X) \cong G(A \otimes_C X) \to G(Y)$ (using subscripts to distinguish the tensor products and unit objects in $\mathbf{C}$ and $\mathbf{D}$).
If $\mathbf{C}$ is already semicocartesian, then $F$ is an isomorphism, because the colimit of any functor (in this case, $\mathrm{Hom}_{\mathbf{C}}(- \otimes X,Y):{\mathbf{C}}^{\mathrm{op}} \to \mathbf{Set}$) whose domain has a terminal object is just given by the evaluation at that terminal object.
But does the above construction really give the correct left $2$-adjoint to the inclusion $\mathbf{SMC}_{0} \hookrightarrow \mathbf{SMC}$, making the $2$-category of semicocartesian symmetric monoidal categories a reflective (and coreflective) sub-$2$-category of the $2$-category of symmetric monoidal categories?