$\newcommand{\C}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\F}{\mathcal F} \renewcommand{\H}{\mathcal H} \newcommand{\from}{\colon} \newcommand{\tensor}{\otimes} \require{AMScd}$
Given monoidal categories $\C,\D$ and a strong monoidal functor $\F\from\C\to\D$, we can define a category $\D_{\F}$ (my name) as follows:
The objects of $\D_{\F}$ are the objects of $\D$.
A morphism from an object $A$ to an object $B$ is an equivalence class of pairs $(X, f)$, where $X$ is an object of $\C$, $f\colon \F(X)\tensor A\to B$ is a morphism in $\D$ and we say that $(X,f)$ is equivalent to $(Y,g)$ if there is some object $Z$ of $X$ and morphisms $h\from Z\to X$, $h'\from Z\to Y$ making the following square commute: $$ \begin{CD} \F(Z)\tensor A @>{\F(h)\tensor A}>> \F(X)\tensor A\\ @V\F(h')\tensor AVV @VVfV \\ \F(Y)\tensor A @>{g}>> B \end{CD} $$
We can define the composition of two morphisms in $\D_{\F}$ using the strong monoidal nature of $\F$: given $(X,f)$ from $A$ to $B$ and $(Y,g)$ from $B$ to $C$, we define the composition of $f$ and $g$ to be the morphism $(Y\tensor X, f;g)$, where $f;g$ is given by the following composite:
\begin{align} \F(Y\tensor X)\tensor A &\xrightarrow{sm} (\F(Y)\tensor \F(X))\tensor A \\&\xrightarrow{\text{assoc}}\F(Y)\tensor (\F(X)\tensor A) \\&\xrightarrow{\F(Y)\tensor f} \F(Y)\tensor B\xrightarrow{g} C \end{align}
This can be shown to be associative (up to the equivalence relation defined above) and so $\D_{\F}$ is a category.
This construction is 2-functorial in the sense that if we have two functors $\F\from \C\to\D$, $\F'\from\C'\to\D'$ and oplax monoidal functors $\H_1\from \C'\to\C,\H_2\from\D'\to\D$ making the obvious square commute, then we can form an oplax monoidal functor $\H\from \D'_{\F'}\to\D_{\F}$ that sends a morphism $(X,f)$ from $A$ to $B$ in $\D'_{\F'}$ to the morphism $(\H_1(X), \H f)$ from $\H_2(A)$ to $\H_2(B)$ in $\D_\F$, where $\H f$ is the following composite in $\D$: \begin{align} \F(\H_1(X)) \tensor \H_2(A) & = \H_2(\F'(X))\tensor \H_2(A)\\ &\xrightarrow{om} \H_2(\F'(X)\tensor A)\xrightarrow{f} \H_2(B) \end{align}
Moreover, this construction respects composition of functors.
Lastly, we can perform a similar construction with natural transformations: given a pair of monoidal natural transformations \begin{gather} \phi_1\from \H'_1\to\H_1\\ \phi_2\from \H'_2\to\H_2 \end{gather} commuting with everything in sight, we can form a monoidal natural transformation between the functors $\H'$ and $\H$ as constructed above. We end up with a 2-functor $$ \mathbf{Arr_{SM}}(\mathbf{MonCat})\to\mathbf{MonCat} $$ where $\mathbf{MonCat}$ is the 2-category of monoidal categories, oplax monoidal functors and monoidal natural transformations and $\mathbf{Arr_{SM}}(\mathbf{MonCat})$ denotes the subcategory of its arrow category consisting of all those objects $\C\xrightarrow{\F}\D$ where the functor $\F$ is strong monoidal.
My question: has this functor a name, and can anyone point me towards a good reference for this construction?
