Has anyone heard of a Lax Monoidal Functor where one of the arrows is flipped?

Question

I've encountered the following scenario

I have a category $\mathcal C$ and for every object $c\in\mathcal C$, I have found a monoidal category $(\mathcal D, \otimes_c, 1_c,\cdots)$ such that I'm able to functorially map $j:c\to c'$ to:

1. $\epsilon_j\,:1_{c'}\to 1_c$;
2. $\mu^j: \otimes_c\Rightarrow\otimes_{c'}$.

Note that $\mathcal D$ doesn't vary, so effectively, I could think of this setting as $$ \operatorname{id} = F : \mathcal D\to\mathcal D $$ $$ \epsilon_j : 1_{c'} \to F(1_c) $$ $$ \mu^j: F\circ\otimes_c \Rightarrow F\otimes_{c'} F $$

Now, because my brain is so smooth it could be a half-decent manifold, I read $\mu^j$ as being $F\otimes_{c'} F\to F\circ\otimes_{c}$ (because, remember, $F$ is the identity, it disappears, so the arrow was simply flipped 😞) which would make that data a candidate for a lax monoidal functor. And I was very happy about that (because they all happened to be injections of some sort, etc. I was sure it was going to work) for a while until I started going over my notes. What I wanted was a (pseudo)functor from $\mathcal C$ to the 2-category of monoidal categories and lax monoidal functors.

I know for a fact that it is possible to, essentially, invert some of the arrows of the monoidal functor coherence diagrams and get diagrams using the funky $\mu^j$, although it messes with the unitality squares a bit. For any reader's convenience, these are the “inverted diagrams” I was thinking about: Take $F:(\mathcal A,1,\otimes,\cdots)\to (\mathcal B,1,\otimes,\cdots)$, any good old functor between the underlying categories; $\epsilon:1_\mathcal B\to F(1_\mathcal A)$; and lastly $\mu:F\circ\otimes_\mathcal A\Rightarrow F\otimes_\mathcal B F$. Let's call that data a “foobar monoidal functor” if they jointly satisfy

$\require{AMScd}\require{mathtools}$ \begin{CD} F(x) @>F(\lambda^{-1})>> F(1\otimes x)\\ @V \lambda^{-1} V V @VV \mu V\\ 1 \otimes F(x) @>>\epsilon\otimes F(x)> F(1)\otimes F(x) \end{CD} \begin{CD} F(x) @>F(\rho^{-1})>> F(x\otimes 1)\\ @V \rho^{-1} V V @VV \mu V\\ F(x)\otimes 1 @>>\epsilon\otimes F(x)> F(x)\otimes F(1) \end{CD} $$\array{ F((x\otimes y)\otimes z) & \xrightarrow{F\alpha} & F(x\otimes (y\otimes z))\\ \mu\downarrow & & \downarrow\mu\\ F(x\otimes y)\otimes F(z) & & F(x)\otimes F(y\otimes z)\\ \hspace{-3.2em}\mu\otimes F(z)\downarrow & & \downarrow\alpha\\ (F(x)\otimes F(y))\otimes F(z) & \xrightarrow{\alpha} & F(x)\otimes (F(y)\otimes F(z)) }$$

Question:

• Is there a name for “foobar” functors?
  • are they common/interesting/important/has anyone ever even seen anything like them?
• How best to go about proving that they are the morphisms of a (2-)category (if they indeed are)? They do form a category with the obvious composition as I've now verified.

Thank you for reading, any help/non-null pointers will be greatly appreciated.