Given a small category $\mathsf{C}$, a presheaf is another name for a contravariant set-valued functor on $\mathsf{C}$. A Grothendieck topos is a reflective full subcategory $\mathsf{E}$ of a presheaf category \(\xymatrix{ \cE \ar@<-1ex>@{^(->}[r] \ar@{}[r]\mid-\perp & \textup{\textsf{cat}}^{\mathsf{C}^\mathrm{op}} \ar@<-1ex>[l]_-L}\) with the property that the left adjoint preserves finite limits. Objects in $\mathsf{E}$ can be characterized as sheaves on a small \emph{site}, which specifies a ``covering family’’ of morphisms $(f_i \colon U_i \to U)i$ for each object $U \in \mathsf{C}$ satisfying a weak pullback condition. A typical example might take $\mathsf{C} =\mathcal{O}(X)$ to be the poset of open sets for a topological space $X$. A presheaf $P \colon \mathcal{O}(X)^\mathrm{op} \to \textup{\textsf{cat}}$ assigns a set $P(U)$ to each open set $U \subset X$ so that this assignment is functorial with respect to restrictions along inclusion $V \subset U \subset X$ of open subsets. A presheaf $P$ is a sheaf if and only if the diagram of restriction maps \(\xymatrix{ P(U) \ar[r] & \prod\limits_\alpha P(U_\alpha) \ar@<.5ex>[r] \ar@<-.5ex>[r] & \prod\limits_{\alpha,\beta} P(U_\alpha \cap U_\beta)}\) is an equalizer for every open cover $U = \cup\alpha U_\alpha$ of a $U \in \mathcal{O}(X)$; see Definition \ref{defn:sheaf-axiom}.