We have an exact sequence of sheaves:
0 $\to$ F' $\to$ F $\to$ F'' $\to$ 0
Then we have exactness of 0 $\to$ F'|U $\to$ F|U $\to$ F''|U $\to$ 0 for an open U of X as exactness of sheaves is stalkwise.
I don't quite follow the last statement from P382 of Rotman's Intro to Homological Algebra because he states on P301 that an exact sequence in pSh implies the its exactness in Sh, but the converse is not true.
In other words, I don't quite see how being exact on stalks implies the exact sequence: 0 $\to$ F'|U $\to$ F|U $\to$ F''|U $\to$ 0. The converse is true, I am aware.
Any enlightenment would be appreciated!