I am currently try to understand some notes on topological and differentiable stacks by J. Heineloth. Here, the pullback of stacks is defined at definition 2.1. Let $\mathcal{M} \stackrel{F}{\to} \mathcal{N} \stackrel{F'}{\leftarrow} \mathcal{M}'$ be a diagram of stacks over a category $\mathcal{C}$. Then, we define the pullback of this diagram as the following stack (at least these are the objects of the groupoid at $Y \in \mathcal{C}$) $$ (\mathcal{M}\times_\mathcal{N}\mathcal{M}')(Y) := \{(m, \alpha, m')\mid m \in \mathcal{M}(Y),m' \in \mathcal{M}'(Y), \alpha : F(m) \simeq F(m')\},$$ which satisfies the obvious universal property. Using the Yoneda embedding, we can view $\mathcal{C}$, as a subcategory of stacks. (Assuming the Grothendieck topology on $\mathcal{C}$ is subcanonical.) Now, if we let $\mathcal{M}:= C \in \mathcal{C}$ and $\mathcal{M'} := C'$ in the above pullback construction, we can view $F$ as an element of $\mathcal{N}(C)$ and $F'$ as an element of $\mathcal{N}(C')$ by the Yoneda lemma. Now, it is claimed in Heineloth page 9 that $C \times_\mathcal{N} C'$ represents the isomorphisms between $F$ and $F'$. In particular, they seem to claim that \begin{equation} (C \times_\mathcal{N} C')(Y) \cong \{f : Y \to C \times C', \alpha : F\pi \simeq F'\pi'\}, \tag{1}\end{equation} where $\pi : C \times C' \to C$ and $\pi' : C \times C' \to C'$ are the canonical projections. This indeed looks like an object that represents isomorphisms between $F$ and $F'$. In particular, we see that the objects of this stack are generated by $f = id_{C \times C'}$ + the isomorphisms between $F$ and $F'$. However, the above description of the pullback does not seem to coincide with the definition. Based on the definition of the pullback of stacks, I would expect that $$(C \times_\mathcal{N} C')(Y) = \{(f_1 : Y \to C, \alpha : Ff_1 \cong Ff_2, f_2 : Y \to C')\}.\tag{2}$$ How can one see that the two objects (1) and (2) are isomorphic?
My struggle lies in producing a map from (2) to (1). I think the first step is that given $f_1 : Y \to C$ and $f_2 : Y \to C'$, we can produce $(f_1, f_2) : Y \to C \times C'$. Furthermore, we may write
$$
\alpha : F\pi_1(f_1, f_2) \cong F\pi_2(f_1, f_2).
$$
Now, in case, $(f_1, f_2)$ is a covering map from $Y \to C \times C'$, I think that we can argue by the fact that $(f_1, f_2)^* : \mathcal{N}(C \times C') \to \mathcal{N}(Y)$ is an equivalence that there is a unique $\beta$ such that $(f_1, f_2)^*(\beta) = \alpha$. However, in general, $(f_1, f_2)$ is not covering, so we cannot use this argument.