Proof.
Proof of (1). We may work locally on $\mathcal{C}$. Hence by Lemma 18.43.3 we may assume we are given a finite diagram of finite sets such that our diagram of sheaves is the associated diagram of constant sheaves. Then we just take the limit or colimit in the category of sets and take the associated constant sheaf. Some details omitted.
To prove (2) and (3) we use the criterion of Homology, Lemma 12.10.3. Existence of kernels and cokernels is argued in the same way as above. Of course, the reason for using a Noetherian ring in (3) is to assure us that the kernel of a map of finite $\Lambda $-modules is a finite $\Lambda $-module. To see that the category is closed under extensions (in the case of sheaves $\Lambda $-modules), assume given an extension of sheaves of $\Lambda $-modules
\[ 0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0 \]
on $\mathcal{C}$ with $\mathcal{F}$, $\mathcal{G}$ finite type and locally constant. Localizing on $\mathcal{C}$ we may assume $\mathcal{F}$ and $\mathcal{G}$ are constant, i.e., we get
\[ 0 \to \underline{M} \to \mathcal{E} \to \underline{N} \to 0 \]
for some $\Lambda $-modules $M, N$. Choose generators $y_1, \ldots , y_ m$ of $N$, so that we get a short exact sequence $0 \to K \to \Lambda ^{\oplus m} \to N \to 0$ of $\Lambda $-modules. Localizing further we may assume $y_ j$ lifts to a section $s_ j$ of $\mathcal{E}$. Thus we see that $\mathcal{E}$ is a pushout as in the following diagram
\[ \xymatrix{ 0 \ar[r] & \underline{K} \ar[d] \ar[r] & \underline{\Lambda ^{\oplus m}} \ar[d] \ar[r] & \underline{N} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \underline{M} \ar[r] & \mathcal{E} \ar[r] & \underline{N} \ar[r] & 0 } \]
By Lemma 18.43.3 again (and the fact that $K$ is a finite $\Lambda $-module as $\Lambda $ is Noetherian) we see that the map $\underline{K} \to \underline{M}$ is locally constant, hence we conclude.
$\square$
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