With the new name wrapper format introduced in biblatex
v3.12 (2018-10-30), Audrey's answer can be simplified quite a bit. We no longer need to redefine the whole labelname
name format, we just append 's
with the name wrapper.
\documentclass{article}
\usepackage[style=authoryear-comp]{biblatex}
\DeclareNameWrapperFormat{labelname:poss}{#1's}
\DeclareFieldFormat{shorthand:poss}{%
\ifnameundef{labelname}{#1's}{#1}}
\DeclareFieldFormat{citetitle:poss}{\mkbibemph{#1}'s}
\DeclareFieldFormat{label:poss}{#1's}
\newrobustcmd*{\posscitealias}{%
\AtNextCite{%
\DeclareNameWrapperAlias{labelname}{labelname:poss}%
\DeclareFieldAlias{shorthand}{shorthand:poss}%
\DeclareFieldAlias{citetitle}{citetitle:poss}%
\DeclareFieldAlias{label}{label:poss}}}
\newrobustcmd*{\posscite}{%
\posscitealias%
\textcite}
\newrobustcmd*{\Posscite}{\bibsentence\posscite}
\newrobustcmd*{\posscites}{%
\posscitealias%
\textcites}
\begin{filecontents}{\jobname.bib}
@article{bertram:sa,
author = {Bertram, Aaron and Wentworth, Richard},
shortauthor = {BW},
title = {Gromov invariants for holomorphic maps on Riemann surfaces},
journaltitle = {J.~Amer. Math. Soc.},
volume = {9},
number = {2},
date = {1996},
pages = {529--571},
}
@book{title,
title = {A Book with No Author},
date = {1993},
}
@book{shorthand,
title = {A Book with No Author},
shorthand = {ABWNA},
date = {1993},
}
@book{label,
label = {Fallback label},
date = {1993},
}
@book{does,
title = {A Book by the Does},
author = {Doe, John and Doe, Jane},
shortauthor = {{The Does}},
date = {1990},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\bibliography{biblatex-examples}
\renewcommand{\baselinestretch}{1.25}
\begin{document}
\subsection*{cite with labelname}
posscite many authors: \posscite[3036]{aksin}
textcite shortauthor: \textcite[e.g.][3]{bertram:sa}
posscite shortauthor: \posscite[e.g.][4]{bertram:sa}
textcite shorthand: \textcite{kant:ku}
posscite shorthand: \posscite{kant:ku}
textcite compact: \textcite{aristotle:physics,aristotle:poetics}
posscite compact: \posscite{aristotle:physics,aristotle:poetics}
textcites: \textcites[3036]{aksin}[529]{bertram:sa}
posscites: \posscites[3036]{aksin}[529]{bertram:sa}
posscite with plural labelname (where this solution is bad): \posscite{does}
\subsection*{cite without labelname}
textcite shorthand: \textcite{shorthand}
posscite shorthand: \posscite{shorthand}
textcite title: \textcite{title}
posscite title: \posscite{title}
textcite label: \textcite{label}
posscite label: \posscite{label}
\end{document}

\citeauthor{texbook}'s \citeyear{texbook}
should work?