Use the counit from part b for the first map and the restriction map $\mathcal G\to i_*i^{-1}\mathcal G$ for the second. Exactness can be checked on stalks. At $P\in U$ the sequence is
$$
0\longrightarrow\mathcal G_P\xrightarrow{1}\mathcal G_P\longrightarrow0,
$$
whereas at $P\in Z$ it is
$$
0\longrightarrow0\longrightarrow\mathcal G_P\xrightarrow{1}\mathcal G_P\longrightarrow0.
$$
Thus
$$
0\longrightarrow j_!j^{-1}\mathcal G\longrightarrow\mathcal G\longrightarrow i_*i^{-1}\mathcal G\longrightarrow0
$$
is a <short exact sequence of sheaves>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 1 / c / Solution

A sequence $0\to\mathcal F'\to\mathcal F\to\mathcal F''\to0$ of sheaves is exact exactly when the induced sequence on every stalk is exact.


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 Incoming links: Short exact sequence of sheaves