category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A general monoidal category $(C,\otimes)$ does not admit diagonal natural transformations of the form $x \longrightarrow x\otimes x$, unlike the case of a cartesian monoidal category (where the monoidal product is the Cartesian product). A monoidal category with diagonals is a monoidal category with the extra structure of a consistent system of such diagonal morphisms.
A consistent system of diagonal maps $\Delta_x\colon x \to x \otimes x$ as $x$ varies through the objects of a monoidal category $(C,\otimes,I)$ should be natural, so that $(f \otimes f)\circ \Delta_x = \Delta_y \circ f$, for any $f\colon x\to y$. Hence such a system is a natural transformation from the identity functor on $C$ to the composite $C \to C \times C \stackrel{\otimes}{\to} C$ of the diagonal functor with the given monoidal product functor.
Another desirable property is that the diagonal map $\Delta_I\colon I \to I\otimes I$ on the tensor unit $I$ is the inverse of the left unitor $\ell_I\colon I \otimes I \stackrel{\sim}{\to} I$ (which is the same as the right unitor $r_I$).
Any cartesian monoidal category has a canonical structure of diagonal maps given by the actual diagonal morphisms.
Given a cartesian monoidal category, the wide subcategory consisting of the monomorphisms is a monoidal category with the same product and unit. The diagonal morphism of the original category, always a monomorphism, plays the role of the diagonal maps in the subcategory. Now there are no projections, since, for instance, in Set, the projection maps are almost never injective.
Write
for the category of pointed topological spaces (with respect to some convenient category of topological spaces such as compactly generated topological spaces or D-topological spaces)
regarded as a symmetric monoidal category with tensor product the smash product and unit the 0-sphere $S^0 \,=\, \ast_+$.
This category also has a Cartesian product, given on pointed spaces $X_i = (\mathcal{X}_i, x_i)$ with underlying $\mathcal{X}_i \in TopologicalSpaces$ by
But since this smash product is a non-trivial quotient of the Cartesian product
it is not itself cartesian, but just symmetric monoidal.
However, via the quotienting (3), it still inherits, from the diagonal morphisms on underlying topological spaces
a suitable notion of monoidal diagonals:
[Smash monoidal diagonals]
For $X \,\in\, PointedTopologicalSpaces$, let $D_X \;\colon\; X \longrightarrow X \wedge X$ be the composite
of the Cartesian diagonal morphism (2) with the coprojection onto the defining quotient space (3).
It is immediate that:
The smash monoidal diagonal $D$ (Def. ) makes the symmetric monoidal category (1) of pointed topological spaces with smash product a monoidal category with diagonals, in that
$D$ is a natural transformation;
$S^0 \overset{\;\;D_{S^0}\;\;}{\longrightarrow} S^0 \wedge S^0$ is an isomorphism.
While elementary in itself, this has the following profound consequence:
[Suspension spectra have diagonals]
Since the suspension spectrum-functor
is a strong monoidal functor from pointed topological spaces (1) to any standard category of highly structured spectra (by this Prop.) it follows that suspension spectra have monoidal diagonals, in the form of natural transformations
to their respective symmetric smash product of spectra.
For example, given a Whitehead-generalized cohomology theory $\widetilde E$ represented by a ring spectrum
the smash-monoidal diagonal structure (5) on suspension spectra serves to define the cup product $(-)\cup (-)$ in the corresponding multiplicative cohomology theory structure:
The stronger notion of relevance monoidal category is discussed in
When a premonoidal category comes equipped with a morphism $\Delta_x\colon x\to x\otimes x$ for all $x$, such as in the Kleisli category for a strong monad on a cartesian category, or in any Freyd category, then the $f$ for which $(f\otimes f)\circ \Delta_x = \Delta_y \circ f$ are called “copyable”.