We　prove　the　Auslander-Reiten　formula　for　the　category　of　fi　nitely　copresented　comodules　over　a　right　semiperfect　coalgebra.　Based　on　this　formula,　we　give　a　su　ffi　cient　condition　for　a　comodule　to　be　partial　tilting.　As　an　application,　we　investigate　the　partial　tilting　and　tilting　homological　properties　of　the　Morita-Takeuchi　context　coalgebra.