I cant understand comments that you implied and is written in the paper.I dont understand what here mean by fibre evaluation and why we have that surjection for first map.

I wrote.these things are in FP-notes.You mean more refrences?

Can you introduce some references? I am not familiar with these things.thanks

I just cant understand why pull_back of basis can split to sections. I dont have problem with rest of the proof.

yes exactly.in fact we can use this with valutative criterion to prove that $\overline M$$_g,_n$$(X,\beta)$ is separated.

Oh,Sorry.So i didnt understand you correctly.

I agree with you that deformation of graph of a morphism in product of domain and target is same as deformation of morphism with domain and target held fixed and it is Def(f).but it is different with $Def_R(f)$.As i understand you described Def(f) not $Def_R(f)$ Am i right?

But what does (C held rigid) means?

I am reading Fulton-panharipande note and they write this exact sequence.