algebra.

In this paper, we generalize Kuperberg’s construction to the context of involutory Hopf algebras in arbitrary symmetric monoidal categories where the tensor products of structural morphisms are contracted with morphisms which, in general, are different from integrals and cointegrals (at least when the latter are not two-sided). More precisely, we define the notion of a *good pair* of an involutory Hopf algebra in a symmetric monoidal category and prove that such a pair gives rise to a topological invariant of closed oriented 3-manifolds (see Theorem \[main-thm\]). Also, we give a way to derive examples of good pairs from integrals and cointegrals when the distinguished grouplike elements are central (see Theorem \[thm-gp\]). In the particular case where the integrals and cointegrals are two-sided (i.e., the distinguished grouplike elements and the distinguished algebra morphisms are trivial), the associated good pair is given by the integral and cointegral themselves, reproducing thereby Kuperberg’s original construction.

The paper is organized as follows. In Section \[sec1\], we describe the notation and conventions used and specify our setup which is given by a symmetric monoidal category, together with Hopf algebras and invertible objects in it. In Section \[sec2\], based on Kuperberg’s work [@Ku1], first we recall 