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We prove two identities that connect some natural tensor products in thecategory $\sf{LCS}$ of locally convex spaces with the tensor products in thecategory $\sf{Ste}$ of stereotype spaces. In particular, we give sufficientconditions under which the identity $$ X^\vartriangle\odot Y^\vartriangle\cong(X^\vartriangle\cdot Y^\vartriangle)^\vartriangle\cong (X\cdot Y)^\vartriangle$$ holds, where $\odot$ is the injective tensor product in the category$\sf{Ste}$, $\cdot$, the primary tensor product in the category $\sf{LCS}$, and$\vartriangle$, the pseudosaturation operation in the category $\sf{LCS}$.Studying the relations of this type is justified by the fact that they turn outto be important instruments for constructing duality theory based on the notionof envelope. In particular, they are used in the construction of the dualitytheory for the class of (not necessarily, Abelian) countable discrete groups.

On tensor fractions and tensor products in the category of stereotype spaces