I have been trying to learn some deformation theory, and came across the following in a paper:


The first order deformations of a morphism of smooth curves $f:X\rightarrow Y$ is in bijection with $H^0(X,f^\*(\mathcal{T}\_Y))$.


I would like to understand a proof of this. I understand some simple facts, like $H^1(X,\mathcal{T}\_X)$ being bijective with the first order deformations of $X$. The reference given was to a paper of Ravi Vakil's, but I am unfamiliar with anything but the very basics of stacks. Does anyone know of a reference for this fact that doesn't use stacks?



Reference Request: Deformations of a map bijective to global sections of the pullback of the tangent sheaf
There is actually the following general result.


Let us consider a morphism of algebraic schemes $f \colon X \to Y$, where $X$ is reduced and projective and $Y$ smooth. Then the first order deformations of $f$ *leaving the domain and the target fixed* are parametrized by $H^0(X, f^\* T\_Y)$.


For a proof (that does not use stacks!) see Sernesi' book "Deformations of algebraic schemes", in particular Proposition 3.4.2 page 158. 


