by looking at structure-preserving transformations between them. The *morphisms* between sheaves provide this context. Most authors tend to focus on the class of morphisms between sheaves over the same space, though for our purposes is quite essential to study sheaf morphisms involving different spaces.

([@TSPBook] or [@Bredon Sec. I.4]) \[df:morphism\] Suppose that $\shf{R}$ is a sheaf on a poset $Y$ and that $\shf{S}$ is a sheaf on $X$. A *sheaf morphism* $m: \shf{R}\to \shf{S}$ along an order preserving map $f: X \to Y$ (careful: $m$ and $f$ *go in opposite directions!*) consists of a set of functions $m_x: \shf{R}(f(x)) \to \shf{S}(x)$ for each $x \in X$ such that the following diagram commutes $$\xymatrix{
      \shf{R}(f(y)) \ar[r]^{m_y} &\shf{S}(y)\\
      \shf{R}(f(x)) \ar[r]_{m_x} \ar[u]^{\shf{R}(f(x) \le f(y))}&\shf{S}(x) \ar[u]_{\shf{S}(x \le y)}\\
      }$$ for each $x \le y$. We usually call the functions $m_x$ the *components* of the sheaf morphism. A sheaf morphism is said to be *injective* (or *surjective*) if each component is injective (or surjective).

\[prop:morphism\] A sheaf morphism $m:\shf{R}\to\shf{S}$ along an order preserving $f:X \to Y$ induces a function taking global sections of $\shf{R}$ (a sheaf over $Y$) to global sections of $\shf{S}$ (a sheaf over $X$).

Suppose $r$ is a section of $\shf{R}$. If 