The Stacks project

Remark 48.30.6. Let $(A_ n)$ and $(B_ n)$ be inverse systems of a category $\mathcal{C}$. Let us say a linear-pro-morphism from $(A_ n)$ to $(B_ n)$ is given by a compatible family of morphisms $\varphi _ n : A_{cn + d} \to B_ n$ for all $n \geq 1$ for some fixed integers $c, d \geq 1$. We'll say $(\varphi _ n : A_{cn + d} \to B_ n)$ and $(\psi _ n : A_{c'n + d'} \to B_ n)$ determine the same morphism if there exist $c'' \geq \max (c, c')$ and $d'' \geq \max (d, d')$ such that the two induced morphisms $A_{c'' n + d''} \to B_ n$ are the same for all $n$. It seems likely that Deligne systems $(K_ n)$ with given value on $U$ are well defined up to linear-pro-isomorphisms. If we ever need this we will carefully formulate and prove this here.