Opmonoidal functor

A functor with natural maps $F_2:F(X\otimes Y)\to FX\otimes FY$ and $F_0:FI\to I$ satisfying the reversed monoidal functor associativity and unit diagrams. No invertibility is required.

A natural transformation $\tau :F\Rightarrow G$ satisfying $G_2{\tau }_{X\otimes Y}=({\tau }_X\otimes {\tau }_Y)F_2$ and $G_0{\tau }_I=F_0$.