Regular immersion and canonical exact sequences between conormal sheaves

Question

Let $f:X \rightarrow Y$, $g:Y \rightarrow Z$ be regular immersion of locally noetherian . One can then show that $g \circ f$ is a regular immersion. In the book I am reading it is stated that we have a canonical exact sequence $$0 \rightarrow f^\ast \mathcal{C}_{Y/Z} \rightarrow \mathcal{C}_{X/Z} \rightarrow \mathcal{C}_{X/Y} \rightarrow 0$$ Where $\mathcal{C}$ is the conormal sheaf of the regular immersion. I am curious - how is the first map in this exact sequence defined? I know that The Stacks Project has a proof of it, but it seems very overkill and messy, so I wonder whether there nice ways to define this map locally.

Answer 1

Vakil has an explanation (with pictures!) of the affine version which I think is very nice: http://math.stanford.edu/~vakil/0708-216/216class50.pdf