If \mathbb {C} = \mathsf {Set} regarded as a discrete 2-category, as noted, a pseudomonoid action is just a monoid action in the ordinary sense---that is, a monoid M, a set X and a function m \cdot x so that m \cdot (n \cdot x) = (mn) \cdot x. The para construction is then the action category, whose objects are the points of X and whose morphisms x \to y are elements m so that m \cdot x = y (with multiplication as composition).

Let \mathsf {CartSp} denote the category of Cartesian spaces \mathbb {R}^n and smooth maps between them, which acts on itself by Cartesian product. Then there is a functor \mathsf {SmMfd} \to \mathsf {Optic}(\mathsf {SmMfd}) (= \mathsf {Lens}(\mathsf {SmMfd})) which carries \mathbb {R}^n to the pair {\mathbb {R}^n \choose \mathbb {R}^n} and a smooth map f: \mathbb {R}^n \to \mathbb {R}^m to the pair (f, Rf) where Rf: \mathbb {R}^n \times \R ^m \to \mathbb {R}^n is the reverse derivative, where Rf(x,-) : \mathbb {R}^m \to \mathbb {R}^n is the linear map given by the transpose of the Jacobian.

Moreover this is monoidal for the Hancock tensor on \mathsf {Optic}(\mathsf {CartSp}). Since \mathsf {Para}(-) is functorial, we obtain a functor \mathsf {Para}(\mathsf {CartSp}) \to \mathsf {Para}(\mathsf {Optic}(\mathsf {CartSp})). Given some parametrized (smooth) map X \times P \to Y, we obtain an optic {X \choose X} \otimes {P \choose P} \to {Y \choose Y}. After choosing x \in X and a section d: Y \to Y---for example, by sampling x,y^* from a dataset and letting d(y) be the gradient of (y-y^*)^2 at y---the resulting gradient P \to P describes how to update the parameters to move y in the desired direction. The fact that this is functorial is essentially the mechanism behind the backpropagation algorithm---see Reference [backprop-as-functor], Reference [bruno-etal-categorical-learning-2021] for more.

(Note that this construction can also be generalized to smooth manifolds, but one must pass to dependent lenses and use the cotangent manifold instead.)

If a category has coproducts, then \mathsf {Set} acts on it via S \cdot X = \coprod _{s \in S}X. (This is the \mathsf {Set}-enriched case of what in enriched categories is called a copower or tensor, the dual of the power objects from Example [efr-K3YI]). The morphisms of \mathsf {Para}_\mathsf {Set}(\mathcal {C}) are pairs (I \in \mathsf {Set}, (f_i: X \to Y \in \mathcal {C})_{i \in I}), which compose in the obvious way.

Note that this definition clearly makes sense even if \mathcal {C} does not actually have coproducts. This is an example of another construction which has been called \mathsf {Para}, which takes a monoidal category \mathcal {V} and a \mathcal {V}-enriched category \mathcal {C} and constructs a double category where the morphisms are pairs (J \in \mathcal {V}, J \to \mathcal {C}(X,Y)). It is not hard to see that this also extends to a double category in the same way, but we do not presently know the correct definition of internal enriched object that would replace pseudomonoid actions to replicate our general theory for this case. (note that the literature contains a notion of internal enriched category, Reference [internal-enriched], but these are categories enriched over internal monoidal categories---that is, the ambient category \mathcal {E} is a 1-category and the categorical structure of the base of enrichment \mathcal {V} is formulated on top of this, not as part of the structure of the objects of the category \mathcal {E})

Let \mathcal {C} be symmetric monoidal, and let S: \mathcal {C} \to \mathsf {Set} be a symmetric lax monoidal functor. Then the category of elements \int _\mathcal {C} S = \{(X \in \mathcal {C}, s \in S(X))\} acquires a symmetric monoidal structure (with the laxator and unitor giving the multiplication on the elements s), so that the obvious forgetful functor \int _\mathcal {C} S \to \mathcal {C} is symmetric.

With the action \int _\mathcal {C} \curverightarrow \mathcal {C} induced by this functor, the (horizontal) morphisms of \mathsf {\mathbb Para}_{\int \mathcal {C}}(\mathcal {C}) are parametrized morphisms P \otimes X \to Y equipped with an element of S(P), and the 2-cells are reparametrization morphisms which preserve the decoration. This may be denoted \mathsf {\mathbb Para}_S(\mathcal {C})