  We prove two identities that connect some natural tensor products in the
category $\sf{LCS}$ of locally convex spaces with the tensor products in the
category $\sf{Ste}$ of stereotype spaces. In particular, we give sufficient
conditions 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 out
to be important instruments for constructing duality theory based on the notion
of envelope. In particular, they are used in the construction of the duality
theory for the class of (not necessarily, Abelian) countable discrete groups.
