\qquad 
  f^{\dag\dag} = f \, .$$ An arrow $f \colon A \to B$ is *dagger monic* when $f^\dag
\circ f = {\ensuremath{\mathrm{id}_{A}}}$, and a *dagger iso(morphism)* if both $f$ and $f^\dag$ are dagger monics.

A *symmetric monoidal dagger category* is a dagger category with a symmetric monoidal structure $({\ensuremath{\mathbf{D}}}, {\ensuremath{\otimes}}, I, \lambda,
\rho, \alpha, \sigma)$ such that $$(f \otimes g)^\dag = f^\dag \otimes g^\dag$$ and moreover the natural isomorphisms $\lambda$, $\rho$, $\alpha$, $\sigma$ are componentwise dagger isomorphisms.

### Examples {#ex:cats .unnumbered}

-   The category ${{\ensuremath{\mathbf{Hilb}}}}$ of Hilbert spaces and continuous linear maps, and its (full) subcategory ${{\ensuremath{\mathbf{fHilb}}}}$ of finite-dimensional Hilbert spaces. Here the dagger is the adjoint, and the tensor product has its standard interpretation for Hilbert spaces. Dagger isomorphisms are *unitaries*, and dagger monics are *isometries*.

-   The category ${{\ensuremath{\mathbf{Rel}}}}$ of sets and relations. Here the dagger is relational converse, while the monoidal structure is given by the cartesian product. This generalizes to relations valued in a commutative quantale [@rosenthal:quantales], and to the category of relations of any regular category [@carbonikasangianstreet:relations]. This has a full sub-category ${{\ensuremath{\mathbf{fRel}}}}$, of finite sets and relations.

-   The category ${{\ensuremath{\mathbf{lbfRel}}}}$, of *locally bifinite relations*. This is the subcategory of ${{\ensuremath{\mathbf{Rel}}}}$ comprising those relations 