the combinatorics of Heegaard diagrams associated to 3-manifolds and then explain how, by using the structural maps of an involutory Hopf algebra in a symmetric monoidal category, one associates a tensor to each (ordered, oriented, and based) Heegaard diagram. In Section \[sec3\], we introduce the notion of a good pair and prove that such a pair gives rise to a 3-manifold invariant (Theorem \[main-thm\]). Also, we illustrate the general construction with an example of an involutory super Hopf algebra (i.e., an involutory Hopf algebra in the symmetric monoidal category of super vector spaces). In Section \[sec4\], we describe how one can construct good pairs starting from integrals and cointegrals (Theorem \[thm-gp\]).

Algebraic preliminaries {#sec1}
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In this section, ${\mathcal{S}}$ is a symmetric monoidal category, with monoidal product $\otimes$, unit object ${\mathbb{1}}$, and symmetry $\tau=\{\tau_{X,Y} {\colon}X \otimes Y \to Y \otimes X\}_{X,Y \in {\mathrm{Ob}}({\mathcal{S}})}$. Recall that ${\mathrm{End}}_{\mathcal{S}}({\mathbb{1}})$ is a commutative monoid (with unit $1={\mathrm{id}}_{\mathbb{1}}$) which acts on each Hom set by the monoidal product. Since ${\mathcal{S}}$ is symmetric, this action satisfies that $$\alpha (f \otimes g)=(\alpha f) \otimes g=f \otimes (\alpha g)$$ for all morphisms $f,g$ in ${\mathcal{S}}$ and all $\alpha \in {\mathrm{End}}_{\mathcal{S}}({\mathbb{1}})$. The elements of ${\mathrm{End}}_{\mathcal{S}}({\mathbb{1}})$ are called *scalars*. For 