(S⊗-) has a right adjoint, suggesting the filtered structure to impose on function sets: The degree of a map f:S->T would be how far it falls short of being a morphism, sups(degT(f(s))−degS(s)), as this is what makes S⊗U->T one-to-one with U->(S->T).
(S⊗-) has a right adjoint, suggesting the filtered structure to impose on function sets: The degree of a map f:S->T would be how far it falls short of being a morphism, sups(degT(f(s))−degS(s)), as this is what makes S⊗U->T one-to-one with U->(S->T).