I'm looking at the definition of 2-sheaf in the nlab http://ncatlab.org/nlab/show/2-sheaf and I get stuck with the definition of 2-separated. Especially with the expression $μ_{ij}(y)∘X(p_{ij})(b_i)=X(q_{ij})(b_i)∘μ_{ij}(x)$ because $\mu_{ij}$ is a 2-cell while $y$ is just an element of the category $X(u)$, in fact I don't even understand any part of it . It may happen that I just don't know the notations involved. Any explanation would be appreciated.
Best regards
I don't know about the nlab but 2-sheaves are usually called stacks and you run in the higher version of separatedness for presheaves. A presheaf of sets is called separated if we check that sections are equal on covers. In symbols, $a, b \in F(U)$ are such that for a cover $U_i \to U$ we have $a|U_i = b|U_j$ then $a = b$.
If F is groupoid valued then one needs to climb up a rung of the Cech ladder. If $a,b$ are objects of the groupoid $F(U)$ such that there exist isomorphisms $\alpha_i \colon a|U_i \to b|U_i$, whose restrictions moreover coincide on double intersections $\alpha_i|U_{ij} = \alpha_j|U_{ij}$ then there exists an isomorphism $\alpha\colon a \to b$, such that $\alpha|U_i = \alpha_i$.
But how many choices of $\alpha$ do we have? Here we need separatedness for morphisms. (and I guess hence the nlab's distinction between 1- and 2-separatedness. I suggest Vistoli's notes on descent, around page 75, for this stuff) Let $\alpha,\beta\colon a \to b$ be two morphism in the groupoid $F(U)$. If $\alpha|U_i = \beta|U_i$ we want $\alpha = \beta$.
What's missing for F to be a stack is going in the opposite direction (existence rather than uniqueness, one usually says $F$ is effective). For morphisms, you want them to glue like a sheaf, so if $\alpha_i,\beta_i$ are a collection of morphisms on $U_i$ such that $\alpha_i|U_{ij} = \beta_j|U_{ij}$ then you want an $\alpha$ on U such that $\alpha|U_i = \alpha_i$. (and separatedness ensures that it is unique)
For objects you need cocycles: $a_i \in F(U_i)$, together with $\alpha_{ij} a_i|U_{ij} \to a_j|U{ij}$, satisfying the cocycle identity on $U_{ijk}$, there exists $a \in F(U)$ such that $\beta_i : a|U_i \to a_i$, and $\alpha_{ij} = \beta_j\beta_i^{-1}$ on $U_{ij}$. (again by separatedness $a$ is unique up to a unique isomorphism)
Of course one can go on and define $3$-sheaves (2-stacks), but you need go up the ladder and start taking quadruple intersections and so on.
It seems that you need to read the following $X(μ_{ij})(y)∘X(p_{ij})(b_i) = X(q_{ij})(b_i)∘X(μ_{ij})(x)$ where for instance $X(μ_{ij})(y)$ is the component at $y$ of the natural transformation $X(μ_{ij})$.