If a lax (symmetric) monoidal functor F:C-->D is a functor above Fin_* that sends coCartesian lifts of inert morphisms to coCartesian lifts, how should I view a colax (symmetric) monoidal functor? First this would mean its opposite is lax, thus C^op--->D^op sends inert coCartesian lifts to coCartesian lifts. But I am having a hard time to understand what the coCartesian lifts of inert morphisms would look like in C^op

There should be an associated Cartesian morphism C'--->(Fin_*)^op whose fiber over 1 is equivalent to C

@MaximilienPéroux If C^\otimes, D^\otimes over Fin_* are symmetric monoidal infinity-categories, you want to take the fibrewise op to get cocartesian fibrations C^{op,\otimes}, D^{op,\otimes}, which are again symmetric monoidal (since op preserves products - this corresponds to composing the functors to Cat with op : Cat -> Cat).

C^{op,\otimes} is the symmetric monoidal structure on C^op with the same tensor product as in C. Then an oplax symmetric monoidal functor is just a lax symmetric monoidal functor C^{op,\otimes} -> D^{op,\otimes}.

Now if C_{\otimes} -> (Fin_*)^op is the cartesian fibration corresponding to the same functors as C^\otimes, then C^{op,\otimes} = (C_{\otimes})^op so this is the same as a functor C_{\otimes} -> D_{\otimes} that preserves cartesian morphisms over inerts (op) in Fin_*^op

@RuneHaugseng Thanks Rune! I am actually aware on how to dualize cartesian and cocartesian fibrations, let me maybe specify my current problem. Let C and D be O-monoidal ∞-categories. A lax O-monoidal functor C-->D is a functor over O that sends cocartesian lifts over inert morphisms in O to cocartesian lifts. We can write the category Fun^lax(C,D). Also, we have Alg_O(C)=Fun^lax(O, C).

Now as you say for O=Fin_*, we can define O-monoidal structures on C^op and D^op using dualization of cocartesian and cartesian fibrations as well explained by Barwick-Glasman-Nardin. Now an oplax O-monoidal functor F:C-->D should be precisely if its opposite F^op:C^op--->D^op is lax O-monoidal with respect to the O-monoidal structure of C^op and D^op

Then Fun^{oplax}(C, D) should be (Fun^{lax}(C^op, D^op))^op.

Now if define CoAlg_O(C)=(Alg_O(C^op))^op, using again the O-monoidal structure on C^op

I would have thought I would get then Fun^colax(O, C)=coAlg_O(C) but it doesn't seem to work

It seems that with the assumptions I have made, I obtain this equivalence: Fun^{colax}(O^op, C)=coAlg_O(C), but I find that O^op appearing here confusing

(Btw Rune, I love your latest paper)