There are various characterizations for an $R$-module to be projective. Two of them can be generalized to any category:
i) $P$ is an object such that given morphisms $\alpha: A \rightarrow B$ and $\varphi: P \rightarrow B$ such that $\alpha$ is an epimorphism, there exists a morphism $\widetilde{\varphi}: P \rightarrow A$ such that $\alpha \circ \widetilde{\varphi} = \varphi$. (Lifting property) In other words, the functor $\text{Hom}_{\mathsf{C}}(P,-): \mathsf{C} \rightarrow \mathsf{Set}$ preserves epimorphisms.
ii) $P$ is an object such that every epimorphism with codomain $P$ splits.
It is easy to see that (i) $\Rightarrow$ (ii) holds in general, by taking $\varphi = \text{id}_P$. Also if the category has the property that for every object $X$ there is an epimorphism $\pi:P \rightarrow X$ where $P$ satisfies (i) (an abelian category with this property is said to have enough projectives), we can show the reverse implication (ii) $\Rightarrow$ (i).
My first question: Does (ii) $\Rightarrow$ (i) hold in general? I suspect the answer is no. If so, under which conditions does it hold?