It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ to be the category of quasi-coherent sheaves on $U$. Given a morphism $f : U \to V$, we have a functor $f^* : Qcoh(V) \to Qcoh(U)$. However we don't have $(gf)^*=f^*g^*$ on the nose, but there is a canonical natural equivalence between them. See Vistoli's notes Section 3.1 and 3.2.1.
Now let $X$ be a scheme and $U_i$ be an open cover of $X$. We have the following cosimplicial diagram of categories $$ \prod Qcoh(U_i)\rightrightarrows\prod Qcoh(U_i\times_X U_j)\text{triple arrows}\prod Qcoh(U_i\times_X U_j\times_X U_k)\ldots $$ Keep in mind that this diagram is only a pseudo diagram in $Cat$, the $2$-category of categories, i.e. the cosimplicial identities only hold up to canonical natural equivalence.
It is also well-known that the descent data of quasi-coherent sheaves is given by a collection $(\xi_i,\phi_{ij})$, where $\xi_i$ is a quasi-coherent sheaf on each $U_i$ and $\phi_{ij}$ is an isomorphism $pr_2^*\xi_j\to pr_1^*\xi_i$ in $Qcoh(U_{i}\times_X U_j)$ which satisfies the cocycle condition $$ pr_{13}^*\phi_{ik}=pr_{12}^*\phi_{ij}\circ pr_{23}^*\phi_{jk}. $$ on $U_i\times_X U_j\times_X U_k$. See Vistoli's notes Section 4.1.2.
The above descent data is given "by hand". On the other hand, I've heard that descent data is a kind of (homotopy) limit. Nevertheless Vistoli's notes doesn't consider homotopy limit.
$\textbf{My question}$ is: is there any reference which studies the (pseudo) homotopy limit (I'm not sure whether we should call it 2-categorical limit) of the above cosimplicial pseudo-diagram and show that it coincides with the descent data given in the literature?
