In the subcategory of topological vector locally convex spaces Hausdorff are built proper classes of weakly reflective and weakly coreflective subcategories, respectively. We proved that the right product of two subcategories often leads us to a weakly reflective subcategory, which is not reflective. The problem formulated in the paper is answered [2], when the right product is a reflective subcategory. For notations, see [1-2]. Definition. A full subcategory R of category C is called weakly reflective, if for any X ∈ |C| there is an object rX ∈ |R| and a morphism rX : X → rX with the property: for an object A ∈ |R| any morphism f : X → A extends through rX : f = g · rX for un g. If the extension is always unique, then R is called the reflective subcategory. We denote by R (respectively K ) the class of non-zero subcategories of category C2V. For K ∈ K and R ∈ R with the respective functors k : C2V → K and r : C2V → R, either μK = {m ∈ Mono|k(m) ∈ Iso}, εR = {e ∈ Epi|r(e) ∈ Iso}. Both μK and εR are classes of bimorphisms. Definition. 1. Let K ∈ K and R ∈ R be. The subcategory SμK(R) is called the right product of subcategories K and R and it is noted by K ∗d R. 2. The subcategory QεR(K) is called the left product of subcategories K and R and it is noted by K ∗s R. If B is a class of bimorphisms and A a subcategory, then SB(A) (respectively QB(A)) is the full subcategory of all B-subobjects (respectively: B-factorobjects) of the objects of subcategory A. We examine the following two conditions: a) R contains the subcategory S of spaces with weak topology; b) K contains the subcategory ˜M of spaces with Mackey topology. Theorem 1. The equality rX = uX · vX is the ((μK)⊤,μK)factorization of morphism rX. 1∗. The equality is the ((εR), εR⊥)-factorization of morphism kX. 2. SμK(R) is a weak reflective subcategory of the category C2V and vX : X → vX is the weak replique of object X. 2∗. QεR(K) is a weak coreflective subcategory of the category C2V and wX : wX → X is the weak coreplique of object X. 3. SμK(R) is a reflective subcategory if it meets one of the conditions a) or b). 3∗. QεR(K) is a coreflective subcategory if it meets one of conditions a) or b). Theorem 2. Let R ∈ R be, Σ the coreflective subcategory of the spaces with the most powerful locally convex topology and σ : C2V → Σ. The following statements are equivalent: 1. SμΣ(R) is a reflective subcategory of the category C2V.2. S ⊂ R. Theorem 3. Let K ∈ K be, Π the reflective subcategory of the complete spaces with weak topology and π : C2V → Π. The following statements are equivalent: 1. QεΠ(K) is a coreflective subcategory of the category C2V. 2. ˜ M⊂K. Theorem 4. 1. K∗dR is a reflective subcategory iff when there is a coreflective subcategory K0 so that M⊂ K0 and K∗dR = K0∗dR. 2. K ∗s R is a coreflective subcategory iff when there is a coreflective subcategory R0 so that S ⊂ R0 and K ∗s R = K ∗s R0. Examples 1. Let  ∗dR be a reflective subcategory. Then S ⊂ R,  0 = QEp(  ) = ˜M and  ∗dR = ˜ M∗d R = C2V. 1∗. Let K ∗s  be a coreflective subcategory. Then ˜ M ⊂ K, SMp(  ) = S and K ∗s  = K ∗s S = C2V. 2. Let K ∈ K, R ∈ R and K ⊂ ˜M be. Then K ∗d R = ˜ M∗d R. 2∗. Let K ∈ K, R ∈ R and R ⊂ S be. Then K ∗s R = K ∗s S. 3. Let K ∈ K, K0 = QEp (K), T ∈ K and K ⊂ T ⊂ K0 be. Then T ∗d R = K0 ∗d R. 3∗. Let R ∈ R, R0 = SMp (R), U ∈ K and R ⊂ U ⊂ R0 be. Then K ∗s U = K ∗s R0.