Let $C$ be an abelian category. Suppose that $(N_i)_{i\in I}$ is an inverse system of objects in $C$. Under which conditions does the hypothesis that $$\operatorname{Ext}_C^1(M,N_i)=0\quad\forall i\in I\tag{1}$$ imply $$\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=0?\tag{2}$$

The functor $\operatorname{Ext}^1_C$ is the Yoneda $\operatorname{Ext}^1$-functor. So we don't have to worry whether $C$ has enough injectives/projectives. However, I would be satisfied if there are some results when $C$ is the category of (unitary) left modules over a ring $R$ with unity.

In the case that $C$ is the category of (unitary) left modules over a ring $R$ with unity, every object in $C$ is an inverse limit of injective modules due to this paper. Therefore, an object $M\in C$ satisfies $$\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=\lim\limits_\longleftarrow \operatorname{Ext}_C^1(M,N_i)\tag{3}$$ for any inverse system $(N_i)_{i\in I}$ in $C$ iff $\operatorname{Ext}^1_C(M,N)=0$ for all $N\in C$ iff $M$ is projective.

However, I'm not imposing that $(3)$ should be true. I only suppose that $(1)$ is true, and I'd like to know when $(2)$ is also true. Counterexamples in which $(1)$ is true but $(2)$ is false will also be very helpful. Thank you in advance.

Let $C$ be an abelian category. Suppose that $(N_i)_{i\in I}$ is an inverse system of objects in $C$. Under which conditions does the hypothesis that $$\operatorname{Ext}_C^1(M,N_i)=0\quad\forall i\in I\tag{1}$$ imply $$\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=0?\tag{2}$$

The functor $\operatorname{Ext}^1_C$ is the Yoneda $\operatorname{Ext}^1$-functor. So we don't have to worry whether $C$ has enough injectives/projectives. However, I would be satisfied if there are some results when $C$ is the category of (unitary) left modules over a ring $R$ with unity.

In the case that $C$ is the category of (unitary) left modules over a ring $R$ with unity, every object in $C$ is an inverse limit of injective modules due to this paper. Therefore, an object $M\in C$ satisfies $$\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=\lim\limits_\longleftarrow \operatorname{Ext}_C^1(M,N_i)\tag{3}$$ for any inverse system $(N_i)_{i\in I}$ in $C$ iff $\operatorname{Ext}^1_C(M,N)=0$ for all $N\in C$ iff $M$ is projective.

However, I'm not imposing that $(3)$ should be true. I only suppose that $(1)$ is true, and I'd like to know when $(2)$ is also true. Counterexamples in which $(1)$ is true but $(2)$ is false will also be very helpful. Thank you in advance.

An answer to the dual problem below will also be greatly appreciated. If there are cases where $(1')$ is true but $(2')$ is not, I would also like to see examples.

Let $C$ be an abelian category. Suppose that $(N_i)_{i\in I}$ is an directed system of objects in $C$. Under which conditions does the hypothesis that $$\operatorname{Ext}_C^1(N_i,M)=0\quad\forall i\in I\tag{$1'$}$$ imply $$\operatorname{Ext}_C^1\left(\lim\limits_\longrightarrow N_i,M\right)=0?\tag{$2'$}$$