Projective object

Let $𝖢$ be a category. An object $𝑃 \in 𝖢$ is said to be projective iff it has the following (left) lifting property against epimorphisms: For any morphism $𝑓 : 𝑃 \to 𝐵$ and epimorphism $𝑞 : 𝐴 ↠ 𝐵$ , there exists a factorization $¯ 𝑓 : 𝑃 \to 𝐴$ so that $𝑞 ¯ 𝑓 = 𝑓$ . cat