= Solution
An <opmonoidal functor> $F:(\mathcal C,\otimes,I)\to(\mathcal D,\otimes,J)$ consists of a <functor>, a <natural transformation>
$$
F_2(X,Y):F(X\otimes Y)\longrightarrow FX\otimes FY,
$$
and a <morphism> $F_0:FI\to J$. These maps need not be invertible. If $\alpha,\lambda,\rho$ are the respective <associators> and <unitors>, its axioms are
$$
\alpha^{\mathcal D}(F_2(X,Y)\otimes1)F_2(X\otimes Y,Z)=(1\otimes F_2(Y,Z))F_2(X,Y\otimes Z)F(\alpha^{\mathcal C}),
$$
$$
\lambda^{\mathcal D}(F_0\otimes1)F_2(I,X)=F(\lambda^{\mathcal C}),\qquad\rho^{\mathcal D}(1\otimes F_0)F_2(X,I)=F(\rho^{\mathcal C}).
$$
The first equation has domain $F((X\otimes Y)\otimes Z)$ and codomain $FX\otimes(FY\otimes FZ)$, which fixes the direction of every arrow. This is also called a <colax monoidal functor>.
An <opmonoidal natural transformation> $\tau:F\Rightarrow G$ between <opmonoidal functors> is a <natural transformation> satisfying
$$
\boxed{G_2(X,Y)\tau_{X\otimes Y}=(\tau_X\otimes\tau_Y)F_2(X,Y),\qquad G_0\tau_I=F_0.}
$$
\b[It respects both the tensor and unit comparison maps.] These are <commutative diagrams> and a unit <commutative diagram>; the arrow directions are opposite to those for a <lax monoidal functor>.
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