\le q \in P$, there is a function $\dshf{D}(p \le q):\dshf{D}(q)\to\dshf{D}(p)$, called an *extension function* (or just an *extension*), such that

3.  For each triple $p \le q \le r \in P$, $\dshf{D}(p \le r) = \dshf{D}(p \le q) \circ \dshf{D}(q \le r)$.

If either of the conditions (3) above are not satisfied, we call the construction a *diagram* instead of a sheaf.

\[ex:simple\_sheaves\] In Figure \[fig:lattice\_sheaves\], choosing $$\shf{S}(a) = \mathbb{R},\;\shf{S}(b) = \mathbb{R},\; \shf{S}(c) = \mathbb{R},\; \shf{S}(d) = \mathbb{R},$$ with $$\left(\shf{S}(d \le a)\right)(x) = 2x,\;\left(\shf{S}(d \le b)\right)(x) = x,\;\left(\shf{S}(a \le c)\right)(x) = x,\;\left(\shf{S}(b \le c)\right)(x) = 2x,$$ results in a sheaf. On the other hand, $$\left(\shf{S}(d \le a)\right)(x) = x,\;\left(\shf{S}(d \le b)\right)(x) = x,\;\left(\shf{S}(a \le c)\right)(x) = x,\;\left(\shf{S}(b \le c)\right)(x) = -x,$$ is merely a diagram, because the composition of the maps on the left ($d \to a \to c$) is the identity map, while the other composition ($d \to b \to c$) is not.

Encoding a multi-model system as a diagram is a useful exercise, since consistencies and inconsistencies between the component models are thereby formalized. Those elements of the stalks that are mutually consistent across the entire system, formalized as a sheaf, are called *sections*. Sections are what 