Coalgebra after counit for a monad given by adjunction Let $L\dashv R$ be an adjunction and $LR$ the associated comonad, with comultiplication $L\eta R\colon LR\to LRLR$ and counit $\varepsilon\colon\mathrm{id}\to LR$.
A coalgebra for this comonad is a map $a\colon X\to LRX$ such that
$$\require{AMScd}
\begin{CD}
X @>{a}>> LRX;\\
@V{a}VV @VV{LRa}V \\
LRX @>{L\eta R_X}>> LRLRX;
\end{CD}$$
commutes and such that
$$X\overset{a}{\longrightarrow} LRX\overset{\varepsilon_X}{\longrightarrow} X$$
is the identity of $X$.
Maybe the question is super naive, but if we consider the morphism $a\circ\varepsilon_X \colon LRX\to LRX$, does it commutatively fit the above square? For sure it makes the upper triangle to commute, but what about $L\eta R_X\circ a\circ \varepsilon_X$, is this equal to $LRa$? Could you give a proof or a counterexample?

Ok I think it's not true. Just take the free/forgetful adjunction $L\dashv R$ between groups and sets, and the easiest possible coalgebra $a\colon G\to LRG$ mapping $g\mapsto[g]$. Then $L\eta R_G\circ a\circ \varepsilon_X$ maps $[g_1,\dots,g_n]\mapsto [[g_1\cdot...\cdot g_n]]$ while $LRa$ maps $[g_1,...,g_n]\mapsto [[g_1],\dots,[g_n]]$.
