which are image-finite, meaning that each element in the domain is related to only finitely many elements in the codomain, and whose converses are also image-finite. This forms a monoidal dagger subcategory of ${{\ensuremath{\mathbf{Rel}}}}$. It serves as a kind of qualitative approximation of the passage from finite- to infinite-dimensional Hilbert spaces. For example, a set carries a compact structure in ${{\ensuremath{\mathbf{lbfRel}}}}$ if and only if it is finite.

-   A common generalization of ${{\ensuremath{\mathbf{fHilb}}}}$ and ${{\ensuremath{\mathbf{fRel}}}}$ is obtained by forming the category ${{\ensuremath{\mathbf{Mat}}}}(S)$, where $S$ is a commutative semiring with a specified involution [@heunen:hilbcatsembedding]. Objects of ${{\ensuremath{\mathbf{Mat}}}}(S)$ are finite sets, and morphisms are maps , which we think of as ‘$X$ times $Y$ matrices’. Composition is by matrix multiplication, while the dagger is conjugate transpose, where conjugation of a matrix means elementwise application of the involution on $S$. The tensor product of $X$ and $Y$ is given by $X \times Y$, with the action on matrices given by componentwise multiplication, corresponding to the ‘Kronecker product’ of matrices. If we take $S = {\ensuremath{\mathbb{C}}}$, this yields a category equivalent to ${{\ensuremath{\mathbf{fHilb}}}}$, while taking $S$ to be the Boolean semiring $\{0, 1\}$, with trivial involution, gives ${{\ensuremath{\mathbf{fRel}}}}$.

-   An infinitary generalization of 