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?