Let $\mathcal{F}$ be a sheaf on a topological space $X$. I am trying to verify that:
$$\begin{align} \lim_{\overrightarrow{U\ni x}}F(U)=\mathcal{F}_x:=\{(U,s):U \text{ open nbd of }x, s\in \mathcal{F}(U)\}/\sim \end{align}$$ where $\sim$ is the equivalence relation $(U,s)\sim (V,t)$ if and only if there exists a $W\subset U\cap V$ such that $s|_W=t|_W$.
To be specific, in an arbitrary category, I understand that the direct limit (if it exists) is given by a universal property. One then has to construct the object that satisfies this universal property. I am attempting to verify that the said modulo the equivalence relation above satisfies the universal property. I would also like to understand how given the universal property one could come up with this set module the equivalence relation, since it was essentially handed down to me and seems to work by magic.
I have verified that the top part of the commutative diagram:
where $X=\mathcal{F}_x$, $X_i=V$, $X_j=U$, $f_{ij}=\text{res}^V_U$, and $\phi_U(s)=[U,s]$, works out. What I am worried about is defining the unique map $u$, which I will refer to as the $\theta$ going forward. It is clear that $\theta$ must satisfy: $$ \theta\circ \phi_U=\psi_U $$ for all $U\subset X$, so I am tempted to define $\theta$ by: $$\theta([U,s])=\psi_U(s)$$ but then we need to check that this well defined. So let $[U',s']=[U,s]$, then we have that: $$\theta([U',s'])=\psi_{U'}(s)$$ so I need to somehow check that $\psi_{U'}(s')=\psi_{U}(s)$.
I think that part of the data of the direct limit is that $\psi_V=\psi_{U}\circ \text{res}^V_U$ whenever $U\subset V$ (is this true?). So, we know that since $[U',s']=[U,s]$ that there exists a $W\subset U\cap U'$ such that $s'|_W=s|_W$, i.e. $$\text{res}^{U'}_W(s')=\text{res}^U_W(s)$$ So apply $\psi_W$ to both sides and we obtain that: $$ \psi_W\circ\text{res}^{U'}_W(s')=\psi_W\circ \text{res}^U_W(s) $$ hence: $$\psi_{U'}(s')=\psi_U(s)$$ so $\theta$ is well defined, and $\mathcal{F}_x$ satisfies the universal property of the direct limit.
Does that make sense, or have I made a fatal error somewhere? Also, how would one get to the definition of $\mathcal{F}_x$ from the universal property of the direct limit? I was given the definition of the stalk in terms of the set quotient the equivalence relation by my professor, and then attempted to independently verify that it satisfies the direct limit, but how would I have come up with this definition from the universal property?