Map between localizations induces map on underlying modules for Zariski covering

Question

While working through a proof of this paper,¹ at the middle of page 45, the author's claim of a short exact sequence seems to amount to the following problem:

Let $A$ be a commutative ring and let $s,t\in A$ which generate the unit ideal. Suppose also that $M,N$ are modules over $A$.

Suppose that we have maps between the localizations $$ f_s:M_s\rightarrow N_s $$ $$f_t:M_t\rightarrow N_t$$ $$f_{st}:M_{st}\rightarrow N_{st} $$ such that the diagram $\require{AMScd}$ $$\begin{CD} M_s @>f_s > > N_s\\ @V V V @V V V\\ M_{st} @> >f_{st} > N_{st} \end{CD}$$ and the analogous one for $t$ commute.

Does there exist a map $f:M\rightarrow N$ making all the relevant diagrams commute?

I thought that applying the theory of the faithfully flat descent for $A\rightarrow A_{s}\times A_{t}$ could be a good idea here but led me nowhere.

¹Bhargav Bhatt, Jacob Lurie: A Riemann-Hilbert correspondence in positive characteristic, https://arxiv.org/abs/1711.04148

Answer 1

This* follows immediately from the fact that the presheaf $\mathcal{Hom}(\tilde{M},\tilde{N})$ (Hartshorne's notation) on $\mathrm{Spec}(A)$ given by

$$U\mapsto Hom(\tilde{M}|_U,\tilde{N}|_U)$$

is a sheaf in the Zariski topology. In particular, it verifies the gluing condition (given your two commuting square diagrams for $s$ and $t$) for its sections on an open cover $\{U_s,U_t\}$ such as the one in the question.

* I'm talking about the assertion in the question, not the linked paper