The Stacks project

Remark 91.4.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $(X, \mathcal{O}_ X) \to (X'_ i, \mathcal{O}_{X'_ i})$, $i = 1, 2$ be first order thickenings with ideal sheaves $\mathcal{I}_ i$. Let $h : (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2})$ be a morphism of first order thickenings of $(X, \mathcal{O}_ X)$. Picture

Observe that $h^\sharp : \mathcal{O}_{X'_2} \to \mathcal{O}_{X'_1}$ in particular induces an $\mathcal{O}_ X$-module map $\mathcal{I}_2 \to \mathcal{I}_1$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Let $(\mathcal{K}_ i, c_ i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}_ X$-module $\mathcal{K}_ i$ and a map $c_ i : \mathcal{I}_ i \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{K}_ i$. Assume furthermore given a map of $\mathcal{O}_ X$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that

is commutative. Then there is a canonical functoriality

Namely, thinking of all sheaves $\mathcal{O}_ X$, $\mathcal{O}_{X'_ i}$, $\mathcal{F}$, $\mathcal{K}_ i$, etc as sheaves on $X$, we set given $\mathcal{F}'_2$ the sheaf $\mathcal{F}'_1$ equal to the pushout, i.e., fitting into the following diagram of extensions

We omit the construction of the $\mathcal{O}_{X'_1}$-module structure on the pushout (this uses the commutativity of the diagram involving $c_1$ and $c_2$).