I am currently reading through Martin Olsson's "Algebraic Spaces and Stacks" and am getting in a tangle as to how to properly define the Isom sheaf.
By a stack here I mean specifically one over the category of schemes over a fixed base scheme $S$, with the étale topology.
Now the following is my guess as to a sensible definition as it isn't said explicitly in Olsson.
Given two stacks $\mathscr{X}/S$ and $\mathscr{Y}/S$, and two morphisms $u_1,u_2:\mathscr{Y} \to \mathscr{X}$, define the presheaf $\underline{\text{Isom}}(u_1,u_2):(\text{Sch}/\mathscr{Y})^{\text{op}}\to \text{Sets}$ as follows:
Given any $f:U \to \mathscr{Y}$ with $U$ a scheme, we obtain maps $u_1 \circ f, u_2 \circ f:U \to \mathscr{X}$. By 2-Yoneda, these maps correspond to elements of $\mathscr{X}(U)$ (the fibre of $U$), which I will call $f^*u_1$ and $f^*u_2$. Then define $\underline{\text{Isom}}(u_1,u_2)(f:U \to \mathscr{Y}):=\text{Isom}_{\mathscr{X}(U)}(f^*u_1,f^*u_2)$.
Assuming that this is a sensible definition (for instance when $\mathscr{Y}$ is a scheme it agrees with the general notion for a category fibred in groupoids), my next question is whether this presheaf is already a sheaf. My guess is that we'd like $\mathscr{Y}$ to be an algebraic space, as then we have an étale scheme cover and can use descent.