Let $\textbf{Sh}(X)$ denote the category of all (set - valued) sheaves on a topological space $X$.
My question is: Given sheaves $F,G \in \textbf{Sh}(X)$ and morphisms $\varphi : F \to G$ and $\psi : F\to G$, is there a definition in the literature for the equalizer of $\varphi$ and $\psi$? I looked in Mac Lane's book and searched on the stacks project but couldn't find anything. Can we define it as the sheaf $E$ such that for any $W \subseteq X$ open,
$$E(W) \to F(W) \begin{array}{c} \longrightarrow \\ \longrightarrow \end{array} G(W) $$
is an equalizer?
The sheafification functor $a : PSh(X) \to Sh(X)$ is left exact (meaning it preserves fintie limits). Its right adjoint is the inclusion $i: Sh(X) \to PSh(X)$ of the category of sheaves into the category of sheaves
Being a right adjoint, $i$ preserves all limits. This means that if $E$ is the equalizer of two arrows $f$ and $g$ in $Sh(X)$, then $i(E)$ is the equalizer of $i(f)$ and $i(g)$ in $PSh(X)$.
In $PSh(X)$, limits and colimits are computed pointwise; that is, if $X$ is the (co)limit of $X_j$, then $X(U)$ is the (co)limit of $X_j(U)$
So these facts together tell us how to compute equalizers, or any limit of sheaves.
Colimits are a little harder; $a$, being a left adjoint, preserves all colimits. So we can compute colimits by taking colimits pointwise, then sheafifying:
$$ \text{colim } X_j \cong \text{colim } ai(X_j) \cong a\left( \text{colim } i(X_j) \right) $$