2-natural transformation?
symmetric monoidal (∞,1)-category of spectra
Just as the notion of a monad in a bicategory $K$ generalizes that of a monoid in a monoidal category, modules over monoids generalize easily to modules over monads.
Modules over monads, especially in Cat, are also often called algebras for the monad; see below.
Let $K$ be a bicategory and $t \colon a \to a$ a monad in $K$ with structure 2-cells $\mu \colon t t \Rightarrow t$ and $\eta \colon 1_a \Rightarrow t$. Then a left $t$-module is given by a 1-cell $x \colon b \to a$ and a 2-cell $\lambda \colon t x \Rightarrow x$, where
commute. Similarly, a right $t$-module is given by a 1-cell $y \colon a \to c$ and a 2-cell $\rho \colon y t \Rightarrow y$, with commuting diagrams as above with $y$ on the left instead of $x$ on the right.
Clearly, a right $t$-module in $K$ is the same thing as a left $t$-module in $K^{\mathrm{op}}$. A left $t$-comodule or coalgebra is then a left $t$-module in $K^{\mathrm{co}}$, and a right $t$-comodule is a left $t$-module in $K^{\mathrm{coop}}$.
A $t$-module of any of these sorts is a fortiori an algebra over the underlying endomorphism $t$.
Given monads $s$ on $b$ and $t$ on $a$, an $s,t$-bimodule is given by a 1-cell $x\colon b \to a$, together with the structures of a right $s$-module $\rho \colon x s \Rightarrow x$ and a left $t$-module $\lambda \colon t x \Rightarrow x$ that are compatible in the sense that the diagram
commutes. Such a bimodule may be written as $x \colon s ⇸ t$.
A morphism of left $t$-modules $(x,\lambda) \to (x',\lambda')$ is given by a 2-cell $\alpha \colon x \Rightarrow x'$ such that $\lambda' \circ t\alpha = \alpha \circ \lambda$. Similarly, a morphism of right $t$-modules $(y,\rho) \to (y',\rho')$ is $\beta \colon y \Rightarrow y'$ such that $\rho' \circ \alpha s = \alpha \circ \rho$. A morphism of bimodules $(x,\lambda,\rho) \to (x',\lambda',\rho')$ is given by $\alpha \colon x \Rightarrow x'$ that is a morphism of both left and right modules.
More abstractly, the monads $s$ and $t$ in $K$ give rise to ordinary monads $s^*$ and $t_*$ on the hom-category $K(b,a)$, by pre- and post-composition. The associativity isomorphism of $K$ then gives rise to an invertible distributive law between these, so that the composite $s^* t_* \cong t_* s^* \colon x \mapsto t x s$ is again a monad. Then the category $Mod_K(s,t)$ of bimodules from $s$ to $t$ is the ordinary Eilenberg--Moore category $K(b,a)^{s^* t_*}$.
If $K = Cat$ and $(T,\eta,\mu)$ is a monad on a category $C$, then a left $T$-module $A \colon C \to 1 \to C$, where $1$ is the terminal category, is usually called a $T$-algebra: it is given by an object $A \in C$ together with a morphism $\alpha \colon T A \to A$, such that
and
commute.
In particular, every algebra over a monad $(T,\eta,\mu)$ in $Cat$ has the structure of an algebra over the underlying endofunctor $T$.
$T$-algebras can also be defined as left modules over $T$ qua monoid in $End(C)$. There the object $A$ is represented by the constant endofunctor at $A$.
The Eilenberg-Moore category of $T$ is the category of these algebras. It has a universal property that allows the notion of Eilenberg-Moore object to be defined in any bicategory.
Given bimodules $x' \colon r ⇸ s$ and $x \colon s ⇸ t$, where $r,s,t$ are monads on $c,b,a$ respectively, we may be able to form the tensor product $x \otimes_s x' \colon r ⇸ t$ just as in the case of bimodules over rings. If the hom-categories of the bicategory $K$ have reflexive coequalizers that are preserved by composition on both sides, then the tensor product is given by the reflexive coequalizer in $K(c,a)$
where the parallel arrows are the two induced actions $\rho x'$ and $x \lambda$ on $s$. Indeed, under the hypothesis on $K$ the forgetful functor $Mod_K(r,t) = K(c,a)^{r^* t_*} \to K(c,a)$ reflects reflexive coequalizers, because the monad $r^* t_*$ preserves them, and so $x \otimes_s x'$ is an $r,t$-bimodule.
If $K$ satisfies the above conditions then there is a bicategory $Mod(K)$ consisting of monads, bimodules and bimodule morphisms in $K$. The identity module on a monad $t$ is $t$ itself, and the unit and associativity conditions follow from the universal property of the above coequalizer. There is a lax forgetful functor $Mod(K) \to K$, with comparison morphisms $1_a \to t$ the unit of $t$, and $x x' \to x \otimes_s x'$ the coequalizer map.
If $K = Span(Set)$, the bicategory of spans of sets, then a monad in $K$ is precisely a small category. Then $Mod(K) = Prof$, the category of small categories, profunctors and natural transformations.
More generally, $Mod(Span(C))$, for $C$ any category with coequalizers and pullbacks that preserve them, consists of internal categories in $C$, together with internal profunctors between them and transformations between those.
algebra over a monad, algebra over an endofunctor, coalgebra over an endofunctor, algebra over a profunctor
Discussion of model category structures on categories of coalgebras over comonads is in