We introduce three new categories in which their objects are T-approximation spaces and they are denoted by NTAprS, RNTAprS, and LNTAprS. We verify the existence or nonexistence of products and coproducts in these three categories and characterized theirs epimorphisms and monomorphisms. We discuss equalizer and coequalizer of a pair of morphisms in the three categories. We introduce the notion of idempotent approximation space, and we show that idempotent approximation spaces and right upper natural transformations form a category, which is denoted by RNTApr2S. Let CS be the category of all closure spaces and closure preserving mappings. We define a functor F from RNTApr2S to CS and show that F is a full functor and every object of CS has a corefiection along F.