Consider the sheaf $N$ over a topological space $X$ which sends $U$ to the set of continuous functions $U \to \mathbb{N}$ where $\mathbb{N}$ has the discrete topology and $N(V \subseteq U)(f) = f|_V$.
I think/hope that a sheaf morphism $\mu: N \to Y$ is completely determined by the values $\mu_U$ takes on constant maps. For any $g \in N(U)$, we have that it is the amalgamation of $g|_{g^{-1}(n)}$ indexed by the open cover $(g^{-1}(n))_{n \in \mathbb{N}}$ of $U$. As such, it is completely determined by contant maps. I feel it should be possible and even simple to write $\mu_U(g)$ as an amalgamation of the image of those constant maps, but have not succeeded. Is there a way to do this?
I think that the closest you can get is the following. The sheaf $N$ in your question is the sheaf associated to a presheaf $\Delta_{\mathbb{N}}$ of constant $\mathbb{N}$-valued functions. Now for every presheaf $P$ and a sheaf $\mathcal{F}$ there is a bijection
$$\mathrm{Mor}_{\mathrm{presheaf}}\left(P,\mathcal{F}\right) \cong \mathrm{Mor}_{\mathrm{sheaf}}\left(P^+,\mathcal{F}\right)$$
where $P^+$ is the sheaf associated to $P$ and the bijection is induced by the canonical morphism $i_P:P\rightarrow P^+$. You can view this bijection as a certain statement concerning amalgamations. In your special case you obtain
$$\mathrm{Mor}_{\mathrm{presheaf}}\left(\Delta_{\mathbb{N}},\mathcal{F}\right) \cong \mathrm{Mor}_{\mathrm{sheaf}}\left(\mathcal{N},\mathcal{F}\right)$$
induced by the canonical inclusion $i:\Delta_{\mathbb{N}}\rightarrow \mathcal{N}$.