Consider a presheaf $F:C^{op}\to Sets$. I want to "see" it as an $\infty$ sheaf, i.e., in the notations of Lurie, Higher Topos Theory, as an element of $Fun(N(C)^{op}, \infty-Grpd)$, where $N(C)$ is its nerve as $1$-category.
Now my difficulty is in associating to the functor $F$ an $\infty$-functor with domain $N(C)^{op}$. I am trying to use a "straightening-unstraightening" argument, but I am stuck. I could apply the classical Grothendieck construction to $F$, obtain a (right?) fibration over $C$ and then take nerves, but it feels quite unstandard.
EDIT: Slightly more sophisticated problem. I have a $1$-functor $C\to A$, $C$ a $1$-category, $A$ an infinity category. Is there a standard way to see it as an infinity functor from $N(C)$ to $A$? That is, the identity $1-functors(C, t_1(A))\cong t_1Fun(N(C),A)$ is true? Here $t_1$ is the $1$-truncation.
Any hints? Am I wrong somewhere?
Thank you in advance.
The proposed identity in the edit is very far from true. For instance it would imply the quotient $A\to \tau_1 A$ admits a splitting.
For the first question, a functor $C\to Set$ induces an $\infty$-functor $NC\to NSet$ because $N$ is a functor. And there is the canonical right adjoint $\infty$-functor $NSet\to \infty-Gpd$, with which one may compose. Straightening and unstraightening is a red herring here.