Let $S$ be a semigroup and $X$ a set. A left $S$-action structure on $X$ is a semigroup arrow $S\to \mathsf{Set}(X,X)$.
Let $M$ be an abelian group and $R$ a ring. A left $R$-module structure on $M$ is a ring arrow $R\to\mathsf{Ab}(M,M)$.
In both cases we are defining actions of semigroup/monoid objects in a category $\mathsf C$ within the category of internal monoids $\mathsf{Mon}(\mathsf C)$.
Usually, equivariance is defined in the category $\mathsf {C}$ itself using diagrams in $\mathsf {C}$ involving the monoid structure arrows of the acting object. Can we define equivariance in $\mathsf{Mon}(\mathsf {C})$?
For instance, can we define $R$-linear arrows of $R$-modules in the category of rings? Perhaps in the commutative case?
If the ambient category is monoidal closed, the usual equivariance conditions may be expressed with internal homs. This expression does not live in $\mathsf{Mon}(\mathsf C)$, but I thought it's still worth recording.
Let $M$ be an internal monoid and consider actions $M\to \underline{\mathsf{C}}(A,A)$ and $M\to \underline{\mathsf{C}}(B,B)$. Observe any arrow $f:A\to B$ in $\mathsf C$ induces morphisms by pre-composition and post-composition. This arrow is equivariant precisely if the following diagram commutes.
$$\require{AMScd} \begin{CD} M @>>> \underline{\mathsf{C}}(A,A)\\ @VVV @VV{f_\ast}V\\ \underline{\mathsf{C}}(B,B) @>>{f^\ast}> \underline{\mathsf{C}}(A,B) \end{CD}$$
However, this still takes place in the category $\mathsf C$ and not $\mathsf{Mon}(\mathsf C)$ since the bottom right corner has no monoid structure.