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\begin{document}
\title[]{Stacks HW5 - Comma Category and adjunction}\maketitle
\begin{enumerate}

\item  Let $C$ be a site, and let $X$ be an object in $C$. Recall that
  the comma category $C/X$ inherits the structure of a site. Assume
  that $C$ is subcanonical (which means that for every $X \in C$,
  $h_X$ is a sheaf).
  \begin{enumerate}
  \item Show that there is an equivalence of categories between
    $Sh(C/X)$ and $Sh(C)/h_X$.  
  \item Show that $j^*\colon Sh(C)\to
    Sh(C)/h_X$, given by $F\mapsto (F\times h_X\xrightarrow{p_2}h_X)$
    commutes with finite limits and has a right adjoint
    $j_*$. (Describe $j_*$ explicitly.) 
  \end{enumerate}


\end{enumerate}
\end{document}

