Let $(X,\mathscr O)$ be a locally ringed space, and let $\mathscr F$, $\mathscr G$ be $\mathscr O$-modules. Consider the following facts: $\newcommand{\sF}{\mathscr F}\newcommand{\sG}{\mathscr G}\newcommand{\sO}{\mathscr O}\newcommand{\sHom}{\mathscr Hom}$
Proposition A If $\sF$ is finitely presented and $\sG$ is coherent, then $\sHom_{\sO}(\sF,\sG)$ is a coherent $\sO$-module.
Proposition B If $\sF$ is finitely presented, then $\sHom_\sO(\sF,\sG)_x \to \operatorname{Hom}_{\sO_x}(\sF_x,\sG_x)$ is an isomorphism for all $x\in X$.
I have a minor confusion about the proof of Proposition A. The proof is standard and goes basically as follows:
The question is local, so we may assume that we have an exact sequence $\mathscr O^p \to \mathscr O^q \to \mathscr F \to 0$. Applying $\sHom_\sO(-,\sG)$ yields an exact sequence
\begin{equation}\label{confused}\tag{$\star$} 0 \to \sHom_\sO(\sF,\sG) \to \sHom_\sO(\sO^q, \sG) \to \sHom_\sO(\sO^p, \sG)\end{equation}
which may be identified with $0 \to \sHom_\sO(\sF,\sG) \to \sG^q \to \sG^p$. Since the category of coherent sheaves is closed under direct sums and kernels, $\sHom_\sO(\sF,\sG)$ must be coherent.
My confusion is that both FAC (Paragraph 14, Proposition 6) and EGA I (Chapter 0, Corollary 5.3.7) cite Proposition B in justifying the exactness of $\eqref{confused}$, and I don't see why this is necessary or relevant.