Finally, we are done proving the assertion in this post. In fact the proof of the result is completely mechanical once you know the result for morphisms of schemes (see this post), and once you have developed enough machinery regarding localization of topoi, and ringed topoi. In fact, entirely the same argument is I think going to prove the result for morphisms of DM stacks mentioned briefly in this post but as usual there is the disclaimer that I haven’t worked out the details yet.
I am going to postpone the application of this result to deformations of maps till later, since I first want to start building theory for algebraic stacks. I will start with fixing the two errors in the chapter on groupoids. In fact I know how to fix the errors due to conversations I had, on this blog and by email, with David Rydh and Jarod Alper.