Condition (2) of Definition \ref{definition-coherent} holds for any subsheaf of a coherent sheaf. Thus we get (1). \medskip\noindent Assume the hypotheses of (2). Let us show that $\Ker(\varphi)$ is of finite type. Pick $x \in X$. Choose an open neighbourhood $U$ of $x$ in $X$ such that $\mathcal{F}|_U$ is generated by $s_1, \ldots, s_n$. By Definition \ref{definition-coherent} the kernel $\mathcal{K}$ of the induced map $\bigoplus_{i = 1}^n \mathcal{O}_U \to \mathcal{G}$, $e_i \mapsto \varphi(s_i)$ is of finite type. Hence $\Ker(\varphi)$ which is the image of the composition $\mathcal{K} \to \bigoplus_{i = 1}^n \mathcal{O}_U \to \mathcal{F}$ is of finite type. \medskip\noindent Assume the hypotheses of (3). By (2) the kernel of $\varphi$ is of finite type and hence by (1) it is coherent. \medskip\noindent With the same hypotheses let us show that $\Coker(\varphi)$ is coherent. Since $\mathcal{G}$ is of finite type so is $\Coker(\varphi)$. Let $U \subset X$ be open and let $\overline{s}_i \in \Coker(\varphi)(U)$, $i = 1, \ldots, n$ be sections. We have to show that the kernel of the associated morphism $\overline{\Psi} : \bigoplus_{i = 1}^n \mathcal{O}_U \to \Coker(\varphi)$ has finite type. There exists an open covering of $U$ such that on each open all the sections $\overline{s}_i$ lift to sections $s_i$ of $\mathcal{G}$. Hence we may assume this is the case over $U$. We may in addition assume there are sections $t_j$, $j = 1, \ldots, m$ of $\Im(\varphi)$ over $U$ which generate $\Im(\varphi)$ over $U$. Let $\Phi : \bigoplus_{j = 1}^m \mathcal{O}_U \to \Im(\varphi)$ be defined using $t_j$ and $\Psi : \bigoplus_{j = 1}^m \mathcal{O}_U \oplus \bigoplus_{i = 1}^n \mathcal{O}_U \to \mathcal{G}$ using $t_j$ and $s_i$. Consider the following commutative diagram $$ \xymatrix{ 0 \ar[r] & \bigoplus_{j = 1}^m \mathcal{O}_U \ar[d]_\Phi \ar[r] & \bigoplus_{j = 1}^m \mathcal{O}_U \oplus \bigoplus_{i = 1}^n \mathcal{O}_U \ar[d]_\Psi \ar[r] & \bigoplus_{i = 1}^n \mathcal{O}_U \ar[d]_{\overline{\Psi}} \ar[r] & 0 \\ 0 \ar[r] & \Im(\varphi) \ar[r] & \mathcal{G} \ar[r] & \Coker(\varphi) \ar[r] & 0 } $$ By the snake lemma we get an exact sequence $\Ker(\Psi) \to \Ker(\overline{\Psi}) \to 0$. Since $\Ker(\Psi)$ is a finite type module, we see that $\Ker(\overline{\Psi})$ has finite type. \medskip\noindent Proof of part (4). Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of $\mathcal{O}_X$-modules. By part (3) it suffices to prove that if $\mathcal{F}_1$ and $\mathcal{F}_3$ are coherent so is $\mathcal{F}_2$. By Lemma \ref{lemma-extension-finite-type} we see that $\mathcal{F}_2$ has finite type. Let $s_1, \ldots, s_n$ be finitely many local sections of $\mathcal{F}_2$ defined over a common open $U$ of $X$. We have to show that the module of relations $\mathcal{K}$ between them is of finite type. Consider the following commutative diagram $$ \xymatrix{ 0 \ar[r] & 0 \ar[r] \ar[d] & \bigoplus_{i = 1}^{n} \mathcal{O}_U \ar[r] \ar[d] & \bigoplus_{i = 1}^{n} \mathcal{O}_U \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F}_1 \ar[r] & \mathcal{F}_2 \ar[r] & \mathcal{F}_3 \ar[r] & 0 } $$ with obvious notation. By the snake lemma we get a short exact sequence $0 \to \mathcal{K} \to \mathcal{K}_3 \to \mathcal{F}_1$ where $\mathcal{K}_3$ is the module of relations among the images of the sections $s_i$ in $\mathcal{F}_3$. Since $\mathcal{F}_1$ is coherent we see that $\mathcal{K}$ is the kernel of a map from a finite type module to a coherent module and hence finite type by (2). \medskip\noindent Proof of (5). This follows because (3) and (4) show that Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory} applies.