Let $X$ be a strict $\infty$-category (not $(\infty,1)$, I am talking about true $\infty$-categories (Grothendieck modules (exact presheaves (finite-limit preserving functors $\Theta^{op}\to \mathrm{Set}$) over Joyal's category $\Theta$ (see Dimitri Ara's thesis)). Is there a notion of a Grothendieck fibration between two such $\infty$-categories or a notion of a 1-cell in inside such a strict $\infty$-category being a Grothendieck fibration? If so, how is it formulated?



Fibrations in strict infinity categories?
It depends on whether you want to mimic the 1-categorical case strictly, or take into account the homotopy nature of the $\infty$-categories. Clearly (or I would hope so!) there is an underlying 1-category of an $\infty$-category, and so a trivial way of arriving at a fibration. I guess this is not what you are after. 


My guess would be to imitate the definition along the lines of what I think Street gave in 'Fibrations in bicategories', i.e.


* For every morphism $f: X \to B$, the canonical map $i: f/\_{\cong} p \to f/p$ has a right adjoint in the slice 2-category $K / X$


where $f/\_{\cong} p$ is the 2-pullback, and $f/p$ is the comma object (from the [nLab](http://ncatlab.org/nlab/show/fibration+in+a+2-category)). So we should just replace 2-categorical concepts here by $\infty$-categorical concepts.


This presumes the definition of adjoint $\infty$-functors :)


As to your second question, if we know that $\infty$Cat is enriched over itself, we use the $\infty$-categorical version of the first definition at [fibration in a 2-category](http://ncatlab.org/nlab/show/fibration+in+a+2-category) (nLab again)




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EDIT: Since you are after Grothendieck fibrations, not Street fibrations, you would want the $\infty$-versions of strict 2-pullbacks and strict comma objects. This should be a significant simplification. I"m not sure, but you may have strict $\infty$-pullbacks = strict $\infty$-iso-comma object, and if both the comma- and iso-comma-objects can be constructed using cotensors, then to me this would be the best approach. But it of course depends on what you want to do next (aside: yes, I saw your other post in that other place).


