$\mathrm{\infty }$-cosheafification

Cosheaves are a dual notion of sheaves. In this paper, we prove existence of a dual of sheafifications, called $\mathit{\text{cosheafifications}}$, in the $\mathrm{\infty }$-category theory. We also prove that the $\mathrm{\infty }$-category of $\mathrm{\infty }$-cosheaves is presentable and equivalent to an $\mathrm{\infty }$-category of left adjoint functors.