For example, given etale maps $p : X \to B$, $r : Y \to B$ and continuous maps $f, g : X \to Y$ such that $p = r \circ f = r \circ g$, the equalizer of $f$ and $g$ seems to be $E = \lbrace x \in X \mid f x = g x \rbrace$ with the inclusion map $e : E \to X$ and the etale map $p \circ e : E \to B$. This works because whenever $f$ and $g$ agree at a point, they agree on a neighborhood of the point, and so $E$ is an open subspace of $X$.

Similarly, if I am not mistaken, the coequalizer of $f$ and $g$ is again computed simply as the quotient $Q = Y/{\sim}$ where $\sim$ is the least equivalence relation satisfying $f x \sim g x$ for all $x \in X$. The equivalence relation $\sim$ is fiber-wise, i.e., it never relates things from different fibers. The canonical quotient map $q : Y \to Q$ is compatible with $r$ and so we get a map $t : Q \to B$, which turns out to be etale.

Is this correct? Where can I find a reference? Most textbooks that prove the category of etale maps over a given base to be a topos make a detour via the category of sheaves on $B$, which makes the calculations of limits and colimits indirect, as they pass through the equivalence between etale maps and sheaves.