Let $A$ be a commutative ring with identity. I denote by $\Delta_{\leq 1}$ the full subcategory of the simplex category $\Delta$ with objects $[0]$ and $[1]$. Let $B_{\cdot} : \Delta_{\leq 1}^{\mathrm{opp}} \longrightarrow \mathrm{Alg}_A$ be a $1$-truncated cosimplicial $A$-algebra.
I'm interested in the descent of $B_{\cdot}$-modules to $A$. More precisely, let $\mathrm{Mod}_{B_{\cdot}}^{c}$ be the category of cosimplicial $B_{\cdot}$-modules $M_{\cdot}$ such that for any morphism $f : [n] \rightarrow [m]$ in $\Delta_{\leq 1}$, the map $$ M_n \otimes_{B_n,B(f)} B_m \longrightarrow M_m $$ induced by $M(f)$ is an isomorphism. There is an adjunction $$ a^* \ : \ \mathrm{Mod}_A \leftrightarrows \mathrm{Mod}_{B_{\cdot}}^{c} : a_* $$ given by $(a^* M)_n = M \otimes_A B_n$ and $a_* M_{\cdot} = \lim ( M_0 \rightrightarrows M_1) $.
When are $a_*$ and $a^*$ equivalences of categories ? In other words, when are the unit $1 \rightarrow a_* a^*$ and the counit $a^* a_* \rightarrow 1$ isomorphisms ?
When $B_{\cdot}$ is the $0$-coskeleton of its $0$-skeleton, i.e. when it has the form $B_1 = B_0 \otimes_A B_0$, with the usual morphisms, then there is a nice answer to this question : $a_*$ and $a^*$ are equivalences of categories precisely when $A \rightarrow B_0$ is universally injective (see Stacks 08WE). Is there a similar characterization in general ?
