Question:
I will first state the question, then do explanation of terminology below.
All categories should be considered as cocomplete stable $\infty$-categories, and colimits, limits etc are the corresponding homotopy version. I will use dg category as a model.
Consider the following representation categories of quivers (over field $\mathbb{C}$)
$$ C = Rep_{dg} (\bullet \gets \bullet \to \bullet) , \quad D = Rep_{dg}(\bullet,\bullet)$$ Consider the following pair of adjoint functors $$L: C \leftrightarrow D: R \quad L (a \gets b \to c) = (b, d=ho\varinjlim(a \gets b \to c) ), \quad R(b, d) = (d \gets (b \oplus d) \to d) $$
(Intuitively speaking, $C$ is the rep of a biCartesian square, and map to $D$ is restriction to the upper-left and bottom-right nodes. )
Let $\Omega = LR$ be the comonad acting on $D$. Is the "decomposition" functor $\tilde L: C \to CoMod_\Omega(D)$ an equivalence of category?
I don't know how to tell apart the $\tilde L$-images of the following two objects: $(0 \gets \mathbb{C} \to \mathbb{C})$ and $(\mathbb{C} \gets \mathbb{C} \to 0)$.
Background
(The following review is for the naive categories, not the infinity category version. I don't know what is the correct upgrade to $\infty$-category due to lack of knowledge in $\infty$-categories. I assume one needs to make $\Omega \rightrightarrows \Omega\Omega$ longer into cosimplicial diagram, but don't know how to spell this out.)
Recall certain definitions about comonad. Let $L: C \leftrightarrow D: R$ be a pair of adjoint functors. Let $\Omega = LR: D \to D$ be the associated comonad, with counit map $\epsilon: \Omega \to 1_D$ and comultiplication $\Delta=L(1 \to RL)R: \Omega \to \Omega \Omega$.
Let $CoMod_\Omega(D)$ be the category, whose object is a pair $(x \in D, a: x \to \Omega(x))$, where the co-action morphism $a$ satisfies condition $$ 1_x = \epsilon(x) \circ a: x \to \Omega(x) \to x$$ and that $$ x \to \Omega(x) \rightrightarrows \Omega\Omega(x) $$ is a equalizer, where one arrow comes from $(\Omega \to \Omega \Omega)$ applied to $x$, and another comes from $\Omega$ applied to $x \to \Omega x$.
We say $L$ is comonadic, if the "decomposition" functor $$ C \to CoMod_\Omega(D), \quad c \mapsto (Lc, L(1 \to RL)(c)) $$ is an equivalence of category.
For Barr-Beck (co)monadicity theorem, see Branter's notes. In lecture 2 and 3, you can find the classical version, and in lecture 4 and 5 the $\infty$-category version.