Let $M$ be a pure-injective module. Then $Hom(\varphi,M)$ is surjective for a pure-mono $\varphi$. It is well-known that $\varphi$ is a direct limit of split monos $\varphi_i$ between finitely presented modules. Clearly all $Hom(\varphi_i, M) $ are surjective. This motivates the following question:

Let $M$ be a pure-injective module, $\psi_i$ a directed system of monos between finitely presented modules and $\psi$ the direct limit of this system. If all $Hom(\psi_i, M)$ are surjective, does it follow that $Hom(\psi, M)$ is surjective?

$\DeclareMathOperator\Hom{Hom}$Let $M$ be a pure-injective module. Then $\Hom(\varphi,M)$ is surjective for a pure-mono $\varphi$. It is well-known that $\varphi$ is a direct limit of split monos $\varphi_i$ between finitely presented modules. Clearly all $\Hom(\varphi_i, M) $ are surjective. This motivates the following question:

Let $M$ be a pure-injective module, $\psi_i$ a directed system of monos between finitely presented modules and $\psi$ the direct limit of this system. If all $\Hom(\psi_i, M)$ are surjective, does it follow that $\Hom(\psi, M)$ is surjective?