An analytic category is called a Stone geometry if it is coherent and locally decidable . Any strong mono in a Stone geometry is regular. Since any direct mono is regular, any locally direct mono is regular. Thus any strong mono in a Stone geometry is regular. Any indecomposable object in a locally direct category is simple.
P is an intersection of proper direct monos, therefore is initial.
Thus P is simple. Suppose A is a coherent analytic category. The diagonal map of any finitely copresentable object is locally finitely generated and locally direct, therefore is an intersection of finite direct monos, thus is direct. Suppose P is an indecomposable object. We prove that P is simple.