I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective tensor product $R\overline\otimes R'$ and the projective tensor product $R\operatorname{\hat\otimes} R'.$
A) Do there also exist relative analogues, that is, given two morphisms of topological rings $S\to R$ and $S\to R',$ is there a topological ring, say, $R\operatorname{\hat\otimes}_S R'$?
B) If they do exist, do they satisfy similar universal properties as their algebraic counterpart $R\otimes_S R'$ which is a pushout in the category of commutative rings?
Thank you!