nLab
oplax monoidal functor

If $C$ and $D$ are monoidal categories, an oplax monoidal functor $F:C\to D$ is defined to be a lax monoidal functor $F:C^{\mathrm{op}}\to D^{\mathrm{op}}$. So, among other things, tensor products are preseved up to morphisms of the following sort in $D$:

$${\varphi }_{c,c\prime }:F(c\otimes c\prime )\to F(c)\otimes F(c\prime )$$\phi_{c,c'} : F(c \otimes c') \to F(c) \otimes F(c')

which must satisfy a certain coherence law.

An oplax monoidal functor sends comonoids in $C$ to comonoids in $D$, just as a lax monoidal functor sends monoids in $C$ to monoids in $D$. For this reason an oplax monoidal functor is sometimes called a lax comonoidal functor. The other obvious terms, colax monoidal and lax opmonoidal, also exist (or at least are attested on Wikipedia).

Note that a strong opmonoidal functor –in which the morphisms $\varphi$ are required to be isomorphisms— is the same thing as a strong monoidal functor.