The coalgebraic cover modality $\nabla_{\T}$ for a finitary standard and weak-pullbacks preserving
endofunctor $\T$ of $\Set$ fully captures the modal logic corresponding to $\T$-coalgebras.
We show that, if in addition the functor $\T$ is finitely presentable in the category
of finitary endofunctors of $\Set$, $\nabla_{\T}$ is a relative left adjoint.

Namely, for $\nabla_{\T}$ as a monotone map from the $\T$-lifted preorder $\T\Lang$ of all modal formulas
to the preorder of all modal formulas $\Lang$ there exists a monotone map $g$ from
$\Lang$ to $\Pom\T\Lang$ such that
$$
\nabla_{\T}\alpha\leq b
\mbox{ holds iff }
\alpha\mathrel{\Lift{\leq}}\gamma
\mbox{ for some $\gamma$ in $g(b)$.}
$$
Such monotone maps were called $\OO$-adjoints in~\cite{VS08} and are an instance of a weakened
representability notion relative to a doctrine, see~\cite{karazeris+velebil}.


\begin{thebibliography}{00}
\bibitem{karazeris+velebil} P. Karazeris, J. Velebil: Representability Relative to a Doctrine, \emph{Cah. Topol. G\'{e}om. Diff\'{e}r. Cat\'{e}g.} L-1 (2009), 3--22.
\bibitem{VS08} L. Santocanale, Y. Venema: Completeness for Flat Modal Fixpoint Logics, submitted (December 2008).
\end{thebibliography}