Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$? Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, the notation $\underline{H}^p(X, \mathscr{F})$ is used. (More generally, $\mathscr{F}$ could be an object of the derived category of sheaves on $X$, e.g. some complex of sheaves.) For example we have the expression $H^0(X, \underline{H}^p(X, i_{Z!} i_Z^! \mathscr{K}_X^p))$ appearing right before Lemma 2.4 or $H^0(U, \underline{H}^p(U, i_{Z!}i_Z^! \mathscr{D}_U^{p,\bullet}))$ in the proof of Theorem 3.9.
It seems to be some kind of sheaf, but I cannot find its definition anywhere. Is it the sheaf associated to the presheaf $U \mapsto H^p(U, \mathscr{F})$? Where can I read some basic properties about this construction? I searched in the Stacks Project, in Hartshorne, Lei Fu's book on etale cohomology, Gelfand-Manin's homological algebra, but I couldn't find it.
 A: As indicated in the comments, the notation $\def\HH{\underline{\rm H}}\def\H{{\rm H}}\HH^p(X,F)$ is defined (for example) by Milne in Étale Cohomology as the $p$th right derived functor of the inclusion functor from the category of sheaves of abelian groups to the category of presheaves of abelian groups on the same site.
This is also known as hypercohomology and is studied under this name by Grothendieck, although I could not locate this notation in his paper.
In modern terms, we can compute $\HH^p(X,F)$ by replacing $F$ with a locally quasi-isomorphic presheaf $G$ of chain complexes that satisfies homotopy descent, e.g., by fibrantly replacing in the local injective model structure on presheaves or sheaves of chain complexes, or the local projective model structure on presheaves of chain complexes.
Then $\HH^p(X,F)$ is simply the $p$th cohomology group of $G$ computed in the abelian category of presheaves of abelian groups.
The keyword to look for in older sources is hypercohomology, the newer sources may simply be talking about the derived category of sheaves, and even newer sources might say something like the “cohomology presheaf of an ∞-sheaf of chain complexes”.
