Some categorical aspects of coarse proximity spaces

In this paper, we study some categorical structures of the category $\mathbf{CoarsePro}$, whose objects are coarse proximity spaces and whose morphisms are coarse proximity maps. We investigate the structure of initial, final, embedding and quotient morphisms in the construct $\mathbf{CoarsePro}$. A special attention is paid to investigate quotients by introducing some conditions that they exist. Also, it is shown that bimorphisms are exactly bijective coarse proximity maps, but not isomorphisms. Consequently, $\mathbf{CoarsePro}$ is not balanced.